Differential Galois Theory and Lie-Vessiot Systems - CiteSeerX

Ph.D. Dissertation

Differential Galois Theory and Lie-Vessiot Systems

David Blázquez Sanz Advisor. Juan J. Morales Ruiz

Departament de Matemàtica Aplicada II Universitat Politècnica de Catalunya

Acknowledgements

First of all I want to thank my advisor, Juan José Morales Ruiz. His direction has been decisive to this work. I thank him with all my heart for patiently listening and helping me throughout this time, and encouraging me to continue developing my own approach to differential equations. I also want to thank my teacher at Universidad de Salamanca, Jes´ us Mu˜ noz D´ıaz, for his continuous support, both in academical and personal affairs. In a much more concrete way I have to thank him for helping with the original work of Sophus Lie, and continuously guiding me towards the best point of view. I also want to thank professor Carles Simo. I moved to Barcelona four years ago to work in his dynamical system project. He opened my mind to a wider and less dogmatic perspective of mathematics. I also acknowledge my companions Primitivo B. Acosta Humanez and Sergi Sim´ on i Estrada, for their help and support through these years. I believe that we form an absolutely inhomogenous and wonderful team. There is a big bunch of people who helped me throughout these studies in the most gently way. There is not enough space here, even for list their names. Thus, I will just mention some of them, more or less in the chronological order in which they appeared in my academical path: ´ Arturo Alvarez Vázquez, who encouraged me to research when I was a degree student. He also altruistically presented to me some of his unpublished conclusions that are related with my work. Ricardo Alonso-Blanco, Sonia

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Jimenez and Jes´ us Rodriguez, who shared with J. Mu˜ noz and me the fruitful informal seminar of differential equations at Universidad de Salamanca. Ricardo Pérez-Marco and Kingshook Biswas, who eventually attended the seminar; they encouraged me to study some differential algebra. Arturo Vieiro, Manuel Marcote and Salvador Rodriguez, who supported me during my arrival in Barcelona. Guy Casale, he discussed several points with me in my research and shared his ideas. Alberto Campos, who discussed with me the subject of Lie symmetries and differential invariants. Marlio Paredes, from who I took the formalism of flag manifolds that simplified my results on integrability a lot. Jean-Pierre Ramis, who invited me to Laboratoire Emile Picard, in Toulouse, during the autumn of 2007. He also suggested I study the automorphic systems in the complex analytic context. This idea has been fruitful and contains the germs of many others that I expect to develop in the future. Emmanuel Paul, who also discussed with me the subject of my thesis in Toulouse. Claude Andre Roche, who helped me to find my place and feel comfortable in Toulouse. Hiroyuki Ino, who shared with me a bottle of the best Japanese sake. Teresa Crespo and Zbigniew Hajto, who gently invited me to discuss the topic of strongly normal extensions, and Jerald Kovacic, who discussed with me the algebraic aspects of this thesis during his visit to Barcelona in February of 2008. Miquel Dalmau, Anna Demier, and Claudia French who helped me – fortunately – to write this work in a more than less – hopefully – readable English. I also want to mention my Colombian colleagues from Universidad Sergio Arboleda, Reinaldo Nu˜ nez and Jes´ us Hernando Pérez, who hosted me in Bogotá in the summer of 2006 and again in the summer of 2007. Definitely I want to thank these people, and others, who let me swim –in mathematical research– happy like a fish in crystal waters.

Foreword

Without long practice; one cannot suddenly understand it. Wang Zongyue from Xangxi. It is a pleasure for me to present this thesis. I am satisfied with this work, it gave me an insight into the wheel of the times, together with some mathematical knowledge. It is my feeling that we have contributed at least some new technical points and applications, that I expose here with the aim of attaining the degree of doctor of mathematics. Beyond that, there is nothing new here; everything was already here before me. During performing this work, I have read some of the mathematics that have been written from 19th to 21th century. I could observe how each generation of mathematicians goes back to the topics of their predecessors but from a different point of view. I do not know how deeply each generation is conscious of its debt. It is obvious for us that Ellis Kolchin’s work is reminiscent of Vessiot’s, but he did not make the connection of his theory of strongly normal extensions with Vessiot’s automorphic systems explicit. My work has been much easier to do than theirs, because I can read both; I can see the wheel spinning twice. My aim for this thesis is to put them together. Barcelona, on the seventh day of April, 2008. David Blázquez Sanz

Contents

1 Introduction 2 Complex Analytic Lie-Vessiot Systems 2.1 Introduction to Lie-Vessiot systems . . . . . . . . . . . . . 2.1.1 Non-autonomous Analytic Vector Fields . . . . . . 2.1.2 Superposition Law . . . . . . . . . . . . . . . . . . 2.2 Lie’s Superposition Theorem . . . . . . . . . . . . . . . . 2.2.1 Proof of Local Lie’s Superposition Theorem . . . . 2.2.2 Proof of Global Lie’s Superposition Theorem . . . 2.3 Automorphic Systems . . . . . . . . . . . . . . . . . . . . 2.3.1 Solution Space . . . . . . . . . . . . . . . . . . . . 2.3.2 Logarithmic Derivative . . . . . . . . . . . . . . . . 2.4 Lie’s Reduction Method . . . . . . . . . . . . . . . . . . . 2.4.1 Gauge Transformations . . . . . . . . . . . . . . . 2.4.2 Lie’s Reduction Method . . . . . . . . . . . . . . . 2.5 Analytic Galois Theory . . . . . . . . . . . . . . . . . . . 2.5.1 Galois Bundle . . . . . . . . . . . . . . . . . . . . . 2.5.2 Analytic Galois Bundle and Picard-Vessiot Bundle 2.5.3 Integration by Quadratures . . . . . . . . . . . . . 2.5.4 Infinitesimal Symmetries . . . . . . . . . . . . . . .

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17 17 18 19 21 27 34 40 41 45 47 48 49 52 53 56 59 61

3 Differential Algebraic Geometry 67 3.1 Differential Algebra . . . . . . . . . . . . . . . . . . . . . . . . 67

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CONTENTS

3.2

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3.1.1 Strongly Normal Extensions . . . . . . . . . . . . . 3.1.2 Lie Extensions . . . . . . . . . . . . . . . . . . . . Differential Schemes . . . . . . . . . . . . . . . . . . . . . 3.2.1 Differential Spectra . . . . . . . . . . . . . . . . . . 3.2.2 Differential Schemes . . . . . . . . . . . . . . . . . 3.2.3 Split of Differential Schemes . . . . . . . . . . . . . 3.2.4 Characterization of Strongly Normal Extensions . Schemes with Derivation . . . . . . . . . . . . . . . . . . . 3.3.1 Differential Schemes and Schemes with Derivation 3.3.2 Split of Schemes with Derivation . . . . . . . . . .

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72 74 76 76 80 82 83 85 86 88

4 Galois theory of Algebraic Lie-Vessiot Systems 91 4.1 Differential Algebraic Geometric theory of Lie-Vessiot Systems 91 4.1.1 Differential Algebraic Dynamical Systems . . . . . . . 91 4.1.2 Algebraic Lie-Vessiot Systems . . . . . . . . . . . . . . 94 4.1.3 Algebraicity of Superposition Laws . . . . . . . . . . . 96 4.1.4 Algebraic Logarithmic Derivative . . . . . . . . . . . . 97 4.2 Splitting Field of an Automorphic System . . . . . . . . . . . 102 4.2.1 Action of G(C) on GK . . . . . . . . . . . . . . . . . . 103 4.2.2 Existence and Uniqueness of the Splitting Field . . . . 104 4.2.3 Galois Group . . . . . . . . . . . . . . . . . . . . . . . 109 4.2.4 Galois Correspondence . . . . . . . . . . . . . . . . . . 114 4.2.5 Lie Extension Structure on Intermediate Fields . . . . 117 5 Algebraic Reduction and Integration 5.1 Lie-Kolchin Reduction Method . . . . . . 5.1.1 Vanishing of Galois Cohomology . 5.1.2 Lie-Kolchin Reduction . . . . . . . 5.2 Integrability by Quadratures . . . . . . . 5.2.1 Quadratures in Abelian Groups . . 5.2.2 Liouville and Kolchin Integrability 5.2.3 Linearization . . . . . . . . . . . . 5.3 Integrability of Linear Equations . . . . . 5.3.1 Flag Variety . . . . . . . . . . . . .

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121 121 122 123 129 130 135 138 140 140

CONTENTS

5.3.2 5.3.3

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Automorphic Equations in the General Linear Group 141 Equations in the Special Orthogonal Group . . . . . . 147

6 Conclusions and Work in Progress

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A Algebraic Geometry A.1 Sheaves and Presheaves A.2 Algebraic Varieties . . . A.2.1 Affine Varieties . A.2.2 Schemes . . . . . A.2.3 Functor of Points

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B Algebraic Groups and Homogeneous Spaces B.1 Complex Analytic Theory . . . . . . . . . . . . . B.1.1 Complex Analytic Lie Groups . . . . . . . B.1.2 Complex Analytic Homogeneous Spaces . B.2 Algebraic Groups . . . . . . . . . . . . . . . . . . B.2.1 Composition Law for Non-rational Points B.2.2 Lie Algebra of an Algebraic Group . . . . B.3 Algebraic Homogeneous spaces . . . . . . . . . . B.3.1 Fundamental Fields . . . . . . . . . . . . B.3.2 Basis of Algebraic Homogeneous Spaces .

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Bibliography

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Index

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1 Introduction

The main subject of this thesis is the study of ordinary differential equations that admit superposition laws; we call them Lie-Vessiot systems. These equations are non-autonomous vector fields whose general solution is expressed as a function of a finite number of particular solutions and a finite number of arbitrary constants. They are natural generalizations of systems of linear ordinary differential equations, for whom the principle of linear superposition holds –the general solution is the space of linear combinations of a fundamental system of solutions–. This superposition law means that this class of systems is, in some way, close to integrable –in fact there is a widespread myth between non specialists which says that linear differential equations are integrable– or at least they must have certain mild properties with respect to their integrability.

Historical Outline In the 19th century, Liouville gave us a useful method for the integration of the second order linear differential equation (see [Lv1838]) by means of an elementary remark: the logarithmic derivative v = y ′ /y of the solutions of the equation y ′′ = r(x)y, (1.1) satisfies the Riccati equation v ′ = v 2 − r(x).

(1.2)

He showed how to construct the general solution of (1.1) from a known rational solution of (1.2) – this knowledge probably goes back to Euler but I am not able to point out the right reference. This is one of the first results

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Chapter 1. Introduction

in differential algebra. An important remark on the Riccati equation is that the anharmonic ratio of four particular solutions of (1.2) is constant. Then, through inversion of this constant, we can express the general solution as a function of three particular solutions and an arbitrary constant v(x) = ϕ(v1 (x), v2 (x), v3 (x), λ) = =

v3 (x)(v1 (x) − v2 (x)) − λv1 (x)(v3 (x) − v2 (x)) ; (v1 (x) − v2 (x)) − λ(v3 (x) − v2 (x))

this formula is a superposition law for the Riccati equations.

The general class of equations admitting superposition laws was introduced by the great mathematician S. Lie (see [Lie1888]). They play a special rˆ ole as auxiliary equations in the problem of integrating a differential equation with a known group of symmetries. They soon became a field of interest. Vessiot and Guldberg (see [Ve1893.a] and [Gu1893]) dedicated their attention to this class of equations. Their conceptual proximity to linear equations made them a good candidate for a class of equations in which a complete theory of integrability is possible –in the sense of the Picard-Vessiot theory–. The race then begun to find the characterization of such a class of equations. It was S. Lie who found the solution, published in 1893 (see [Lie1893.b] and [Lie1893.a]). His result is now known as Lie’s superposition or Lie-Scheffers theorem; it relates the equation to a finite dimensional Lie algebra operating in the phase space –nowadays called the Lie-Vessiot-Guldberg algebra–. In Lie’s context it was assumed that this finite dimensional algebra was the infinitesimal generator of a Lie group action in the phase space. Vessiot’s idea was to translate the equation from the phase space to the Lie group, obtaining a particular case of Lie-Vessiot equation he called automorphic because it has the following property: Solutions are related by translations (see [Ve1893.b] and [Ve1894]). Those automorphic equations are conceptually analogous to the linear systems –where the unknown variables are the elements of a fundamental matrix of solutions of a system of ordinary linear differential equations–. Then he tried to generalize Picard-Vessiot theory (see [Ve1892]) to those equations [Ve1904, Ve1940].

State of the Art During the 20th century the theory of Lie-Vessiot systems was developed by two different lines of apparently disconnected mathematicians. First,

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those systems attracted the interest of applied mathematicians because of their frequent appearance in physical problems like stability and orientation of solids. Research in classification (see [Sh-Wi1984]) and applications (see [Sh-Wi1985] and [HPW1999]) was carried out in the eighties. More recently there emerged a branch of mathematicians and physicists who continue this work, and are trying to recover some of the tools and intuitions of S. Lie. In this trend we can take into account the work of J. F. Cari˜ nena, A. Ramos, J. Grabowsky and G. Marmo (see [CGM2000], [CGR2001] [Ca-Ra2002] and [CGM2007]). On the other hand the development of differential algebra occurred. The work of Ritt and Kolchin made a purely algebraic treatment of differential equations possible. Kolchin did not make an approach to Lie-Vessiot systems, but his theory of strongly normal extension is an algebraic counterpart of Vessiot’s theory of automorphic systems. This relation between Kolchin’s theory and Lie-Vessiot was first pointed out by K. Nishioka. Thus, he initiated the algebraic approach to Lie-Vessiot systems and their related differential extensions (see [Ni1989.a], [Ni1989.b] and [Ni1997]). His work is focused on differential equations whose general solution depends algebraically on arbitrary constants. Those differential equations are precisely Lie-Vessiot systems. Finally, we would also like to mention the work of A. Buium (see [Bu1986]), and J. Kovacic (see [Kov2002], [Kov2003] and [Kov2006]) in differential schemes. These works are apparently not focused on Lie-Vessiot systems, but they give us an excellent setting for dealing with them.

Our Approach to Lie-Vessiot Systems The aim of this work is a formalization, in contemporary terms, of original ideas and intuitions shown by S. Lie and E. Vessiot in their works. They led us to a Galois theory for such systems. We found two amazing facts. First, that this theory can be stated in a purely analytic approach; and secondly, that Lie’s algorithms for integration by quadratures fit into our Galois theory and give the results of reducibility and solvability which, in the linear case, were stated by Kolchin. Finally we develop some algebraic tools that allow us to understand the differential extensions which appear when we solve the equations.

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Structure of the Text and Original Results Chapter 2. Complex Analytic Lie-Vessiot Systems In this chapter we approach superposition laws in the complex-analytic context. We analyze the underlying structure of Lie’s superposition theorem. This theorem gives us a partial characterization of the systems admitting superposition laws. Lie stated that a system admits a superposition law if and only if its Lie-Vessiot-Guldberg algebra is finite dimensional. Today, we know that this condition is too weak, partially because it is local. In a recent work Cari˜ nena, Grabowsky and Marmo prove that a finite LieVessiot-Guldberg algebra allows a local superposition law (cf. [CGM2007]). We split then Lie’s theorem into a local and global part. Subsection 2.2.1 is devoted to proving the local Lie theorem as stated above. We do that in order to clarify certain obscure points of the proof as stated in the previous literature, even in the original work of Lie as stated in [Lie1893.b]. The local part is then clarified; for the global part we introduce the notion of pretransitive Lie group action. A relevant result is that transitivity implies pretransitivity; this justifies our terminology. Consider a Lie group action of G in a complex analytic manifold M . Proposition 2.2.1Let M be a faithful homogeneous G-space of finite rank. The action of G in M is pretransitive. ~ whose We define a Lie-Vessiot system as a non-autonomous vector field X Lie-Guldberg-Vessiot algebra is spanned by fundamental fields of a Lie group action in the phase space. Thus, we can state one of our main results. ~ admits a suTheorem 2.2 (Global Lie’s superposition theorem) X perposition law if and only if it is a Lie-Vessiot system related to certain pretransitive Lie group action in M . ~ defined in We focus then our attention on an arbitrary Lie-Vessiot system X an analytic manifold M and for whom the independent variable moves in a Riemann surface S. In order to deal with it, we translate it into a differential equation defined in G, the Lie group acting in the phase space M . This is the so-called automorphic system, and this notion goes back to Vessiot [Ve1904]. This automorphic system induces a hierarchy of Lie-Vessiot systems in the category of G-spaces. We study the principal homogeneous space structure which naturally arises in the space of solutions, and how the solutions of

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systems in the hierarchy are related. We use the logarithmic derivative as a technical tool that allows us to see that gauge transformations are the natural transformations in this hierarchy. We study Lie’s reduction method, and put it together with Grauert theorem on principal bundles. Consider ~ in G, and the related Lie-Vessiot system X ~ in a an automorphic system A, G-homogeneous space M , having as coefficients meromorphic functions in a Riemann surface S. Denote Hx0 for the isotropy group of x0 in M . Then we state: Proposition 2.4.1 Assume that there is a meromorphic solution x(t) of ~ defined in S. Choose t0 ∈ S and denote x0 = x(t0 ). Assume one of the X following additional hypothesis, (a) Hx0 is a special group. (b) S is non compact and Hx0 is connected. In such a case there is a meromorphic gauge transformation in S × G that ~ to an automorphic system in Hx . reduces A 0 Of course, Lie’s reduction method is well known –for an excellent presentation see [Br1991] lecture 3–. The novelty lies in the analytic context, and the remark on the role of global meromorphic transformations under suitable hypothesis. In previous literature, it is always assumed that Lie’s reduction works, but there is not much though on the point that a suitable gauge transformation, defined for all values of the independent variable t, may not exist. Lie’s reduction method leads us to a new purely geometric-analytic approach to differential Galois theory. Our motivation comes from the tannakian formalism as presented J.-P. Ramis and J. Martinet in [RM1990]. We define the ~ (Definition 2.5.2) as a principal Galois bundle of an automorphic system A bundle on S × – where by S × we denote the Riemann surface obtained by ~ –, whose fibers are made up by right removing from S the singularities of A translations that respect the values of meromorphic solutions. This seems to be a particular case of the non-linear Galois theory due to B. Malgrange [Mlg2001, Mlg2002], H. Umemura [Um1996], and developed by G. Casale [Ca2006, Ca2007]. Theorem 2.18 There is a canonical inclusion of the analytic differential Galois group into the algebraic differential Galois group of Picard-Vessiot

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theory. The analytic Galois group is Zariski dense in the algebraic Galois group. This analytic approach is not based on differential algebra, so that there is not a notion of what a Liouvillian extension means. Notwithstanding, we can recover some results analogous to differential algebra by a direct application of Lie’s reduction method. Under suitable hypothesis we have a Kolchin-like reduction to the Lie algebra of the Galois group, and a theorem on integration by quadratures on solvable groups. ~ → S × is contained Theorem 2.19 Assume that the fiber of π : Gal(A) in a connected special group H ⊂ G. Then there is a meromorphic gauge ~ to an automorphic system in H. transform of G × S which reduces A Theorem 2.20 Assume that G is a connected solvable group, and one of the following hypothesis: (a) G is a special group. (b) S is a non-compact Riemann surface. ~ is integrable by quadratures of closed 1-forms Then the automorphic system A in S and the exponential map of G. The last section of chapter two is devoted to the study of infinitesimal symmetries of automorphic and Lie-Vessiot systems. In our case we have a canonical transversal structure, so that we reduce our considerations to the ~ We prove (Theorem 2.21) that sheaf of transversal symmetries, T rans(A). ~ the sheaf of transversal symmetries of A is a free module over the sheaf of ~ which is an extension of the main result of [Ath1997] for first integrals of A, the linear case. We obtain precisely the following: ~ A ~ = L(G) ⊗C OS×G T rans(A) .

In particular, time independent left invariant vector fields are symmetries. However, left invariant vector fields do not act in the homogeneous spaces. We are then interested in symmetries that behave as right invariant vector fields. Those vector fields are solutions of an induced Lie-Vessiot system (equations (2.31)) and form a finite dimensional Lie algebra. We denote ~ the Lie algebra consisting of those symmetries that depend by Right(A) meromorphically on the Riemann surface S. We have a connection with the Galois group:

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~ is contained in the centralizer of Right(A). ~ Theorem 2.23 Gal(A) As a corollary we obtain the following result. It is not explicit in MoralesRamis work [Mo-Ra2001.a, Mo-Ra2001.b] but it is in some sense in the heart of the theory. ~ be an automorphic system in Corollary 2.24 (Morales-Ramis) Let A ~ contains an abelian algebra of the symplectic group Sp(2n, C). If Right(A) 0 ~ dimension n, then Gal (A) is abelian.

Chapter 3. Differential Algebraic Geometry This chapter is devoted to the setting up of the differential-algebraic language that we need in order to develop an algebraic Galois theory for automorphic systems. This chapter is threefold: we consider differential rings and strongly normal extensions, differential schemes, and schemes with derivation. We expose the characterization of strongly normal extensions in terms of differential schemes, due to Kovacic [Kov2003, Kov2006]. Our theoretical goal is the notion of splitting of differential schemes (Definition 3.2.11) and schemes with derivation (Definition 3.3.3). In terms of differential equations, a split scheme with derivation is a differential equation which algebraically reducible to Lie’s canonical form. The main novelty of the chapter is a systematic study of the relationship of differential schemes and schemes with derivation. We do this analysis in order to develop our Galois theory in the frame of schemes with derivation, which is much closer to dynamical geometric intuition than the abstract formalism of differential schemes. To each scheme with derivation (X, ∂) we associate a differential scheme X ′ = Diff(X, ∂) . We have the following result. Theorem 3.13 Given a scheme with derivation (X, ∂), there exists a unique topological subspace X ′ ⊂ X satisfying the following: (1) X ′ endowed with the structure sheaf OX |X ′ and the derivation ∂|X ′ is a differential scheme. This differential scheme will be denoted by Diff(X, ∂). (2) For each open affine subset U ⊂ X, U ∩ X ′ ≃ DiffSpec(OX (U ), ∂).

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Furthermore each morphism of schemes with derivation (X, ∂X ) → (Y, ∂Y ) induces a morphism of differential schemes Diff(X, ∂X ) → Diff(Y, ∂Y ). The assignation (X, ∂) ; Diff(X, ∂) is functorial. This functorial relationship between schemes with derivation and differential schemes allows us to translate the characterization of strongly normal extensions to the language of schemes with derivations (Corollary 3.15). This is useful in our framework, because our automorphic systems are best understood as schemes with derivation.

Chapter 4. Galois theory of Algebraic Lie-Vessiot Systems This chapter is devoted to the development of an algebraic Galois theory of automorphic systems on algebraic groups, hence of Lie-Vessiot systems in algebraic homogeneous spaces. This Galois theory is announced in [Ve1893.b], and is equivalent to the Kolchin’s theory of G-primitive extensions [Ko1973], but this last presentation avoids differential equations. This Galois theory is a generalization of the Picard-Vessiot theory, and should be seen as the natural approach to strongly normal extensions based on differential equations. Consider an algebraically closed field of characteristic zero C, an algebraic group G defined over C, and a G-homogeneous space M . Let K be a differential field with constant field C. A Lie-Vessiot system in M with coefficients in K is a derivation ∂X~ in the extended space MK . Thus, (MK , ∂X~ ) is a scheme with derivations. The same holds for the automorphic system (GK , ∂A~ ). We can also define an automorphic equation (4.2), equivalent to the automorphic system, in terms of the logarithmic derivative: ~ l∂(x) = A.

(4.2)

~ is ~ is the transversal part of the derivation ∂ ~ . We have that A Here, A A an element of R(G) ⊗C K; where R(G) is the Lie-algebra of right invariant vector fields in G. The core of the geometry of algebraic Lie-Vessiot systems is contained in the following result. Theorem 4.9 Assume that the action of G in M is faithful. Then the following are equivalent.

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(1) The automorphic equation (4.2) has a solution with coefficients in K. (2) (GK , ∂A~ ) splits. ~ to 0. (3) There is a gauge transformation of GK sending A (4) (MK , ∂X~ ) splits. (5) (GK , ∂A~ ) splits, is almost constant, and Const(GK , ∂A~ ) ≃ G. (6) (MK , ∂A~ ) splits, is almost constant, and Const(MK , ∂X~ ) ≃ M . Then we see that the problem of solving a Lie-Vessiot system is equivalent to finding a solution for the automorphic equation (4.2). In general, this equation has no solution with coefficients in K. We can then see whether it is possible to extend K to a suitable differential extension K ⊂ L so that the automorphic equation has a solution with coefficients in L. In such case (GL , ∂A~ ) splits. Subsection 4.2.2 is devoted to the existence and uniqueness –up to differential isomorphism– of a smallest splitting extension for the automorphic system, which we call the Galois extension (Definition 4.2.3). We deal with this problem in a constructive way. These Galois extensions are the rational fields of Kolchin closed differential points of (GK , ∂A~ ). The existence is stated in Proposition 4.2.1 (that comes from Lemma 4.11), and the uniqueness up to differential isomorphism is stated in corollary 4.13 of Proposition 4.2.2. Through the analysis of the natural action of G, (GK , ∂A~ ) × G → (GK , ∂A~ ), we obtain a geometrical intrinsic definition for the Galois group Galx(GK , ∂A~ ) (Definition 4.2.4) on a Kolchin closed differential point x of GK . This is an algebraic subgroup of G (Proposition 4.2.5), that depends on x. They all are isomorphic and conjugated by an element of G (Proposition 4.2.6). The Galois group realizes itself as a group of differential automorphisms; hence the Galois extension is a strongly normal extension in the sense of Kolchin. Theorem 4.16 The Galois extension K ⊂ L associated to (GK , ∂A~ ) is a strongly normal extension. ~ is the group of differential K-algebra automorTheorem 4.17 Galx(GK , A) phisms of L. The rest of the chapter is devoted to an analysis of the effects of algebraic group morphisms on automorphic systems, and to the geometrical interpretation of Galois correspondence for strongly normal extensions.

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Chapter 5. Algebraic Reduction and Integration This chapter is devoted to integration by quadratures. The core of the chapter is Lie’s reduction method as presented in chapter two. However, here we deal with the algebraic case. We relate quadratures to differential field extensions that have abelian Galois groups. On the other hand we find certain algebraic obstructions to the applicability of reduction. These obstructions are measured by the Galois cohomology of the group. Because of this kind of algebraic structure behind the reduction we will call this process a Lie-Kolchin reduction. Theorem 5.5 (main result) Let us assume that (MK , ∂X~ ) has a solution x with coefficients in K. Let x0 ∈ M be a rational point and let Hx0 ⊂ G be the isotropy group of x0 . If H 1 (Hx0 , K) is trivial, then there exists a gauge ~ ∈ R(G) ⊗C K to, transformation Lτ of GK sending A ~ = Adjτ (A) ~ + l∂(τ ) B

~ ∈ R(Hx ) ⊗C K. with B 0 We explore different applications of the Lie-Kolchin reduction method. As a corollary we find an extension of Kolchin’s reduction theorem to the theory of automorphic systems. This theorem was stated by Kolchin in the context of Picard-Vessiot theory (linear algebraic groups) and we state the same result for general algebraic groups. Theorem 5.9 (Kolchin) Denote by H the Galois group Galx(GK , ∂A~ ). Assume that the geometric quotient G/H exists. Let us consider the relative algebraic closure K◦ of K in L. Then there is a gauge transformation Lτ , τ with coefficients in K◦ such that, ~ = Adjτ (A) ~ + l∂(τ ), B

is in R(H) ⊗C K◦ . Once we have a reduction we go to the problem of solvability. We analyze the integration by quadratures in abelian and solvable groups. We define Kolchin and Liouville extension as is done in [Ko1953] (Definition 5.2.5) . As a remark we obtain Drach-Kolchin’s theorem (5.10). We translate a result due to Kolchin (Proposition 5.2.1) to the frame of automorphic systems. Theorem 5.11 Assume that there is a chain of resolution of G, H0 H1 . . . Hn = G,

15

such that dimC Hi /Hi+1 = 1. Then the Galois extension K ⊂ L of (GK , ∂A~ ) is a Kolchin extension. In subsection 5.2.3 we give some theoretical results on the linearization of algebraic automorphic systems. The heuristic is evident, we can make use of the results on the structure of algebraic groups to reduce automorphic equations to the linear part. By means of Chevalley-Barsotti-Sancho structure theorem we obtain the following. Theorem 5.13 Consider N the normal linear subgroup of G such that ~ be the induced automorphic system V = G/K is an abelian variety. Let B in V . Let M be the field of meromorphic functions in VK . Assume that Galy(VK , ∂B~ ) = V , and assume one of the following hypothesis: (1) H 1 (N, M) is trivial, (2) K is relatively algebraically closed in L. There exists then a gauge transformation of G with coefficients in M reduc~ to N . ing the automorphic system A Analogously we obtain a reduction by means of the adjoint representation, which is linear. Note that this representation is defined for general Lie groups and hence that the following result is not algebraic. In fact, it is due to Vessiot, and stated in [Ve1893.b]. Proposition 5.2.3 Let K ⊂ P be the Picard-Vessiot extension of the automorphic system in GL(R(G)) induced by (GK , ∂A~ ) and the adjoint representation. Then P ⊂ L is a strongly normal extension with abelian Galois group. Finally we state a result (Proposition 5.2.4) of reduction of the quasi-abelian part of the algebraic group. This result is weaker than the previous one, because the quasi-abelian part of an algebraic group is contained in the kernel of the adjoint representation. In any case, it is interesting because it captures the whole of the linear part of the equation. Once we have the reduction to linear automorphic equations, we develop an automorphic-equation approach to the integrability of linear systems. We obtain that some results concerning the integrability of such equations are best seen under this point of view. As a particular application we see that

16


we can generalize a classical result of Liouville [Lv1838], relating the integrability of second order linear homogeneous equation with the integrability of the Riccati equation. We have that to each algebraic group G it corresponds a complete projective variety F lag(G) and that for an automorphic system the following result holds. Theorem 5.14 The Galois extension K ⊂ L is Liouvillian if and only if the flag Lie-Vessiot system (F lag(G), ∂F~ ) has an algebraic solution with coefficients in K◦ , the algebraic relative closure of K in L. We discuss how Grassmanian manifolds and matrix Riccati equations arise, under this point of view, when we examine the integrability of automorphic equations in the general linear group. In particular, projective Riccati equations are a generalization of the higher order Riccati equations, used by Singer [Si-Ul1993.a, Si-Ul1993.b]. As we expected, algebraic solutions of such equations are related to reduction of order. We compute the general form of such equations, and the general form of the flag equations which contain all the hierarchy of linked matrix Riccati equations associated with the integrability of a general linear system: λ˙ij =

n X k=j

aik λkj −

j X n X k=1 r=j

λik akr λrj +

j j n X X X

λir λrk aks λsj .

(5.11)

k=1 r=k+1 s=j

As a final consideration we see how we can interpret under our point of view a classical result due to Darboux [Da1894] on the equations of rigid motions. We obtain that this result is a particular case of our generalization of Liouville theorem, and then it can be stated in the following form which is slightly more general than Darboux’s original result. Theorem 5.15 (Darboux) The Galois extension of the equation,      x˙ 0 a b x0 x˙ 1  = −a c   x1  a, b, c ∈ K. x˙ 2 −b −c x2

is a Liouvillian extension of K if and only if the Riccati equation, x˙ = has an algebraic solution.

−b + ic 2 −b − ic − iax + x 2 2

2 Complex Analytic Lie-Vessiot Systems

2.1

Introduction to Lie-Vessiot systems

The class of ordinary differential equations admitting fundamental systems of solutions was introduced by S. Lie in 1888 [Lie1888], as certain class of auxiliary equations appearing in his integration methods for ordinary differential equations. Further development, carried out by Guldberg [Gu1893] and Vessiot [Ve1894, Ve1904] relates such equations with differential equations in Lie groups. Final characterization is given by Lie in 1893 [Lie1893.a] - this is the well-known Lie’s superposition theorem. Because of that characterization, these systems are usually referred to as Lie or Lie-Vessiot systems. Further research in classification and applications has been done by Winternitz, Shinner, Cari˜ nena, Grabowsky, Marmo, Ramos and others [Sh-Wi1984], [Sh-Wi1985], [Maa1995],[HRUW1999], [CGM2000], [CGR2001], [Ca-Ra2002], [CGM2007]. An ordinary differential equation admitting a fundamental system of solutions is, by definition, a system of non-autonomous differential equations, dxi = Fi (t, x1 , . . . , xn ) i = 1, . . . , n dt

(2.1)

for which there exists a set of formulae, ϕi (x(1) , . . . , x(r) , λ1 , . . . , λn ) i = 1, . . . , n

(2.2)

expressing the general solution as function of r particular solutions of (2.1) and some arbitrary constants λi . This means that for r particular solutions

18

Chapter 2. Complex Analytic Lie-Vessiot Systems

x(i) (t) of the equations, satisfying certain non-degeneracy condition, xi (t) = ϕi (x(1) (t), . . . , x(r) (t), λ)

(2.3)

is the general solution of the equation (2.1). In [Lie1893.b] Lie also stated that the arbitrary constants λi parameterize the solution space, in the sense that for different constants, we obtain different solutions: there are not functional relations between the arbitrary constants λi . The set of formulae ϕi is usually referred to as a superposition law for solutions of (2.1). Lie’s superposition theorem, as stated in [Lie1893.b], says that a differential equation (2.1) admits a superposition law if and only if the functions Fi can we written in the form Fi (t, x) =

s X

fj (t)ξij (x),

(2.4)

j=1

where the infinitesimal transformations Xi = Ps k k k=1 cij Xk , with cij constants.

Pn

∂ i j=1 ξj (x) ∂xj

verify [Xi , Xj ] =

In contemporary terms, we can say that equation (2.1) admits a superposition law if and only if it has a finite dimensional Lie-Vessiot-Guldberg algebra. In fact, the theorem that was stated by Lie relates a global statement –admitting a superposition law– with a local statement –having a finite dimensional Lie-Vessiot-Guldberg algebra–. As expected, going from the local to the global statement implies the integration of the Lie-Vessiot-Guldberg algebra of the equation. Nowadays it is known that the integration of a Lie algebra of vector fields to the action of a Lie group is not a trivial problem. Moreover, the Lie group action obtained can be highly complicated, chaotic, in the sense that orbits of the action are not embedded into the phase space. For this reason it would be desirable to differentiate global and local aspects of Lie’s superposition theorem. In a recent work [CGM2007], several researchers carried out the analysis of the local aspects of Lie’s superposition theorem. Here we also analyze the global aspect of Lie’s superposition theorem with new arguments, and characterize the class of systems that admit superposition laws in the classical sense of the formula (2.3).

2.1.1

Non-autonomous Analytic Vector Fields

Notation. Along this section M is a complex analytic manifold (phase space), S is a Riemann surface together with a derivation ∂ : OS → OS .

2.1. Introduction to Lie-Vessiot systems

19

By extended phase space we mean the cartesian power S × M . We shall denote t for the general point of S and x for the general point of M . We write x ¯ for a r-frame (x(1) , . . . , x(r) ) ∈ M r . Under this rule, we shall write f (x) for functions in M , f (t) for functions in S and f (t, x) for functions in the extended phase space. Whenever we need we take a local system of coordinates x1 , . . . , xn for M . When we take coordinates in the cartesian (j) power M r we differentiate the components with superindices xi . When computing in the cartesian power M r × M we do not use superindices for coordinates in the last component. ~ in M , Definition 2.1.1 A non-autonomous complex analytic vector field X depending on the Riemann surface S, is an autonomous vector field in S×M , compatible with ∂ in the following sense: ~ (t) = ∂f (t) Xf

~ X

OS×M O

/ OS×M O

∂

OS

/ OS

~ r . This In each cartesian power M r of M we consider the lifted vector field X ~ acting in each component of the cartesian is just the direct sum copies of X r ~ power M . We have the local expression for X, ~ =∂+ X

n X i=1

Fi (t, x)

∂ , ∂xi

~ r , which is a non-autonomous vector field and also the local expression for X in M r , ~r = ∂ + X

n X i=1

2.1.2

(1)

Fi (t, x

)

∂ (1) ∂xi

+ ... +

n X i=1

Fi (t, x(r) )

∂ (r)

∂xi

.

(2.5)

Superposition Law

~ is an analytic map Definition 2.1.2 A superposition law for X ϕ : U × P → M, where U is analytic open subset of M r and P is an n-dimensional manifold, verifying:

20


~r (a) U is union of integral curves of X ~ r , defined for t in some open subset S ′ ⊂ S, (b) If x ¯(t) is a solution of X ~ for t varying in then xλ (t) = ϕ(¯ x(t), λ), is the general solution of X ′ S. Remark 2.1.1 Without any lose of generality we can make the following assumptions for the superposition law: (1) P is isomorphic to M . (2) Considering the above stated isomorphism, for a given t0 in S there is a solution x ¯0 (t), defined for t in a neighborhood of t0 , such that ϕ(¯ x0 (t0 ), x) = x for all x ∈ M . In this case the dependance on arbitrary constants is interpreted as dependance on initial conditions. Let us explain how we make these assumptions. Consider a superposition law φ : U × P → M, ~ r . Then, and choose t0 ∈ S and certain local solution x ¯0 (t) of X x(t, λ) = φ(¯ x0 (t), λ), is the general solution. Define: y : P → M,

λ 7→ x(t0 , λ),

then, by local existence and uniqueness of solutions for differential equations, for each x0 ∈ M there exist a unique local solution x(t) such that x(t0 ) = x0 . Hence, there exist a map ϕ,

Id×y

φ

/M v; v v vv vv ϕ v v

U ×P

U ×M

~ satisfying the above assumptions (1) and which is a superposition law for X (2).

2.2. Lie’s Superposition Theorem

21

Example.[Linear systems] Let us consider a linear system of ordinary differential equations, n

dxi X aij (t)xj , = dt

i = 1, . . . , n

j=1

as we should know, a linear combination of solutions of this system is also a solution. Thus, the solution of the system is a n dimensional vector space, and we can express the global solution as linear combinations of n linearly independent solutions. The superposition law is written down as follows, Cn×n (j) xi

×

Cnλj

n

→C ,

(j) (xi , λj )

7→ (yi )

yi =

n X

(j)

λj x i .

j=1

Example.[Riccati equations] Let us consider the ordinary differential equation, dx = a(t) + b(t)x + c(t)x2 , dt let us consider four different solutions x1 (t), x2 (t), x3 (t), x4 (t). A direct computation gives that the anharmonic ratio is constant, d (x1 − x2 )(x3 − x4 ) = 0. dt (x1 − x4 )(x3 − x2 ) If x1 , x2 , x3 represent three known solutions, we can extract the unknown solution x of the expression, λ=

(x1 − x2 )(x3 − x) (x1 − x)(x3 − x2 )

obtaining, x3 (x1 − x2 ) − λx1 (x3 − x2 ) (x1 − x2 ) − λ(x3 − x2 ) which is the general solution in function of the constant λ, and then a superposition law for the Riccati equation. x=

2.2

Lie’s Superposition Theorem

~ a non-autonomous vector field in M . We can see X ~ Let us consider X ~ as an holomorphic map X : S → X(M ), which assigns to each t0 ∈ S an ~t . autonomous vector field X 0

22


~ is the Lie algebra Definition 2.2.1 The Lie-Vessiot-Guldberg algebra of X ~ t }t ∈S . The Lieof vector fields in M spanned by the set vector fields {X 0 0 ~ ~ Vessiot-Guldberg algebra of X is denoted g(X). ~ is finite dimensional Remark 2.2.1 The Lie-Vessiot-Guldberg algebra of X ~ ~ if and only if there exist X1 , . . . , Xs autonomous vector fields in M , spanning a finite dimensional Lie algebra, and holomorphic functions f1 (t), . . . , fs (t) in S such that, s X ~ i. ~ = fi (t)X X i=1

~ has a finite dimensional Lie-Vessiot-Guldberg algebra if and only Hence, X if it satisfies the condition (2.4).

Thus, having a finite dimensional Lie-Vessiot-Guldberg algebra is the local condition related by Lie’s superposition theorem with local superposition formula. In order to obtain a global superposition formula we have to make a more restrictive assumption. Notation. From now on, let us consider a complex analytic Lie group G, and a faithful analytic action of G on M , G × M → M,

(σ, x) → σ · x.

R(G) is the Lie algebra of right-invariant vector fields in G. We denote R(G, M ) the Lie algebra of fundamental vector fields of the action of G on M (see Appendix B, Definition B.1.6). For each r ∈ N, G acts in the cartesian power M r component by component. There is a minimum r such that there are principal orbits in M r . If M is an homogeneous space then this number r is the rank of M (see Appendix B, Definition B.1.10). For x ¯ ∈ M r , the orbit of x ¯ is denoted Ox¯ and the isotropy subgroup of x ¯, Hx¯ . We say that x is a principal point or that Ox¯ is a principal orbit if Hx¯ is the identity group. In such case Ox¯ is a principal homogeneous G-space. Definition 2.2.2 We say that the action of G is on M pretransitive if there exists r ≥ 1 and an analytic open subset U ⊂ M r such that: (a) U is union of principal orbits.


23

(b) The space of orbits U/G is a complex analytic manifold. Proposition 2.2.1 If M is a G-homogeneous space of finite rank, then the action of G in M is pretransitive. Proof. Let r be the rank of M . The set B ⊂ M r of basis of M is an analytic open subset of M (see Appendix , remark ). It is, by definition, the union of all principal orbits in M r . Let us see that the space of orbits B/G is a complex analytic manifold. In order to that we will construct an atlas for B/G. Let us consider the natural projection: π : B → B/G,

x ¯ 7→ π(¯ x) = Ox¯ .

Let us consider x ¯ ∈ B; let us construct a coordinate open subset for π(¯ x). Ox¯ is a submanifold of B. We take a submanifold L of B such that Ox¯ and L intersect transversally in x ¯, Tx¯ U = Tx¯ Ox¯ ⊕ Tx¯ L. Let us prove that there is an polydisc Ux¯ , open in L and centered on x ¯, r such that Ox¯ ∩ Ux¯ = x ¯. The action of G on M is continuous and free. So that, if the required statement statement holds, then there exists a maybe smaller polydisc Vx¯ centered in x ¯ and open in L such that for all y¯ ∈ Vx¯ the intersection of Oy¯ with Vx¯ is reduced to the only point y¯. Thus, Vx¯ projects one-to-one by π. ∼

π : Vx¯ − → Vπ(¯x)

y¯ 7→ π(¯ y ) = Oy¯

Vπ(¯x) is an open neighborhood of π(x). The inverse of π is an homeomorphism of such open subset of B/G with a complex polydisc. In this way we obtain an open covering of the space of orbits. Transitions functions are the holonomy maps of the foliation whose leaves are the orbits. Thus, transition functions are complex analytic and B/G is a complex analytic manifold. Reasoning by reductio ad absurdum let us assume that there is not a polydisc verifying the required hypothesis. Each neighborhood of x in L intersects then Ox in more than one points. Let us take a topological basis, U1 ⊃ U2 ⊃ . . . ⊃ . . . , of open neighborhoods of x ¯ in L. ∞ \

i=1

Ui = {¯ x},

Ui ∩ Ox¯ = {¯ x, x ¯i , . . .}.

24


In this way we construct a sequence {¯ xi }i∈N in L ∩ Ox¯ such that, lim x ¯i = x ¯,

i→∞

where x ¯i is different from x ¯ for all i ∈ N. The action of G is free, so that, for each i ∈ N there is an unique σi ∈ G such that σi (¯ xi ) = x ¯. We write these r-frames in components, (1)

x ¯ = (x(1) , . . . , x(r) ),

(r)

x ¯i = (xi , . . . , xi ),

we have that for all i ∈ N and k = 1, . . . , r, (k)

x(k) = σi (xi ). Let us denote for all i ∈ N and k = 1, . . . , r, (k)

Hx(k) ,x(k) = {σ ∈ G | σ · xi i

= x(k) },

(k)

the set of all elements of G sending xi to x(k) . This set is the image of the (k) isotropy group Hx(k) by a right translation in G. When xi is closer to x(k) this translation is closer to the identity. We have, σi ⊂

r \

k=1

and from that, lim σi ∈

i→∞

Hx(k) ,x(k) , i

r \

Hx(k) ,

k=1 i→∞

but this intersection in precisely the isotropy group of x ¯ and then σi −−−→ Id.

On the other hand there is a neighborhood of x ¯ in Ox¯ that intersects L only in x ¯. So that there is a neighborhood UId of the identity in G such that UId · x ¯ intersects L only in x ¯. From certain i0 on, σi is in UId , and then σi = Id. Then x ¯i = x ¯, in contradiction with the hypothesis. 2 ~ in M is called Definition 2.2.3 A non-autonomous analytic vector field X a Lie-Vessiot system, relative to the action of G, if its Lie-Vessiot-Guldberg algebra is spanned by fundamental fields of the action of G on M ; if and ~ ⊂ R(G, M ). only if g(X)


25

As we are going to prove, the global counterpart of having finite dimensional Lie-Vessiot-Guldberg algebra is being a Lie-Vessiot system relative to a pretransitive Lie group action. Now, let us discuss the local counterpart of admitting a superposition formula. Let us consider OM the sheaf of holomorphic functions in M , and for each r, the sheaf OM r of holomorphic functions in the cartesian power M r . The ~ r (2.5) define the subsheaf OX~ rr of first integrals of the lifted vector field X M ~ r in M r . vector field X Definition 2.2.4 Let x be a point of M and OM,x be the ring of germs of complex analytic functions in x. A subring R ⊂ OM,x is a regular ring if and only if it is the ring of first integrals of k germs in x of vector fields ~1 ,. . .,Y ~k , linearly which are independent at the point x and in involution: Y ~i , Y ~j ] = 0. [Y The dimension dim R is the number dimC M − k. Consider a sheaf of rings Ψ ⊂ OM r+1 of complex analytic functions in the cartesian power M r+1 . We say that Ψ is a sheaf of generically regular rings if for a generic point x ∈ OM r+1 the stalk Ψx is a regular ring. A sheaf Ψ of generically regular rings is the sheaf of first integrals of a generica1lly regular Frobenius integrable foliation. We denote this foliation by FΨ . ~ is a sheaf of rings Definition 2.2.5 A local superposition law for X Ψ ⊂ OM r+1 , for some r ∈ N, verifying: (1) Ψ is a sheaf of generically regular rings of dimension ≥ n. ~ r+1 Ψ = 0, or equivalently, X ~ is tangent to FΨ . (2) X (3) FΨ is generically transversal to the fibers of the projection M r+1 → M r . This notion is found for first time in [CGM2007], in the language of foliations. ~ is equivalent They prove that the existence of a local superposition law of X to finite dimensional Lie-Vessiot-Guldberg algebra, as we also are going to see. ~ admits a superposition law, then it admits a local Proposition 2.2.2 If X superposition law.

26


~ Proof. Let us consider a superposition law for X, ϕ : U × M → M,

U ⊂ M r,

We write the general solution, x(t) = ϕ(x(1) (t), . . . , x(r) (t), λ).

(2.6)

The local analytic dependency with respect to initial conditions ensures that ∂ϕ1 ,...ϕn the partial jacobian ∂λ does not vanish. Therefore, at least locally, we 1 ,...,λn can invert with respect to the last variables, λ = ψ(x(1) (t), . . . , x(r) (t), x(t)). From that, the components ψi of ψ are first integrals of the lifted vector ~ r+1 . We consider Ψ the sheaf of rings generated by these functions field X ψi . This is a sheaf of regular rings of dimension n. By construction FΨ is transversal to the fibers of the projection M r × M → M . We conclude that ~ this sheaf is a local superposition law for X. 2 Finally we can expose the whole frame of Lie’s superposition theorem splitting its local and global aspects.

Local superposition law Ψ

local Lie’s theorem

integration ~ of g(X)

inversion of Ψ global Lie’s theorem Superposition law ϕ

Finite dimensional LVG algebra

Lie-Vessiot system of pre-transitive action

In the above diagram we differentiate the local statement at the top of the diagram from the global statement in the bottom. It is clear that the global superposition law implies a local superposition law, and that being a LieVessiot system implies finite dimensional Lie-Vessiot-Guldberg algebra. Reciprocal implications are not true in general, and then they are represented by dashed lines arrows in the diagram. We will discuss further this question.


27

Theorem 2.1 (local Lie’s superposition theorem [CGM2007]) The ~ in M admits a local superposition law if non-autonomous vector field X and only its Lie-Vessiot-Guldberg algebra is finite dimensional. Theorem 2.2 (global Lie’s superposition theorem) The non~ in M admits a superposition law if and only autonomous vector field X if it is a Lie-Vessiot system related to certain pre-transitive Lie group action in M .

2.2.1

Proof of Local Lie’s Superposition Theorem

Once we understand the relation between superposition law and local superposition law, we see that Lie’s original proof is still valid. Here we follow [Lie1893.b]. However we detail some points that were not explicitly detailed and remain obscure in the original proof.

Local Superposition Law Implies Finite Dimensional Lie-VessiotGuldberg Algebra ~ admits a local superposition law Ψ in OM r+1 . First, let us assume that X ~ r+1 as a non-autonomous vector field Let us consider the cartesian power X r+1 ~ r+1 . in M . Any section ψ of Ψ is a first integral of X ~ r+1 }t∈S ⊂ X(M r+1 ). Let us take a Consider the family of vector fields {X t r+1 ~ maximal subfamily {X }t∈Λ of OM r+1 -linearly independent vector fields; t here, Λ is some subset of S. The cardinal of a set OM r+1 -linearly independent vector fields is bounded by the dimension of M r+1 . Hence, Λ = {t1 , . . . , tm }, is a finite subset. We denote the corresponding vector fields as follows: ~ r+1 = X ~ r+1 X ti i

i = 1, . . . , m.

We consider the following notation; in the cartesian power M r+1 we denote the different copies of M as follows: M r+1 = M (1) × . . . × M (r) × M.

~ r+1 is the sum, We recall that the lifted vector field X i ~ r+1 = X ~ (1) + . . . + X ~ (r) + X ~ i, X i i i

28


of r + 1 copies of the same vector field acting in the different copies of M . ~ r+1 . Thus, for all i, j, The sheaf Ψ consists of first integrals of the fields X i r+1 r+1 ~ ~ the Lie bracket [X ,X ] also annihilates the sheaf Ψ. We can write the i j Lie bracket as a sum of r + 1 copies of the same vector field in M , ~ r+1 , X ~ r+1 ] = [X ~ (1) , X ~ (1) ] + . . . [X ~ (r) , X ~ (r) ] + [X ~ i, X ~ j ]. [X i j i j i j Let us note that the Lie bracket of two lifted vector fields is again a lifted ~ r+1 span a distribution which is genervector field. The m vector fields X i ically regular of rank m. However, in the general case, it is not Frobenius integrable. We add all the feasible Lie brackets, obtaining an infinite family X =

∞ [

k=0

Xk

where, ~ r+1 , . . . , X ~ r+1 }, X0 = {X m 1 and, ~ , Z] ~ |Y ~ ∈ Xi , Z ~ ∈ Xj for i < k and j < k}. Xk = {[Y The family X is a set of lifted vector fields. They span a generically regular distribution in M r+1 which is, by construction, Frobenius integrable. We extract of this family a maximal subset of OM r+1 -linearly independent vector fields, ~ r+1 , . . . , Y ~sr+1 }. {Y 1 They span the same generically regular Frobenius integrable distribution of rank s ≥ m. Without any lose of generality we can assume that the m vector ~ r+1 are within this family. These vector fields Y ~ r+1 annihilate the fields X i i sheaf Ψ. Thus, we also obtain, s ≤ nr

(2.7)

because the distribution of vector fields annihilating Ψ has rank nr + n − dim(Ψ), and by hypothesis dim(Ψ) is greater than n. ~ r+1 in M r+1 are lifted. We write them as a sum of different The vector fields Y i copies of a vector field in M . ~ r+1 = Y ~ (1) + . . . + Y ~ (r) + Y ~i . Y i i i

(2.8)


29

They span an integrable distribution, so that there exist analytic functions λkij in M r+1 such that, ~ r+1 , Y ~ r+1 ] = [Y i j

s X

~ r+1 , λkij (x(1) , . . . , x(r) , x)Y k

(2.9)

k=1

Let us prove that these functions λkij are, in fact, constants. From (2.8) we have ~ (1) , Y ~ (1) ] + . . . + [Y ~ (r) , Y ~ (r) ] + [Y ~i , Y~j ] ~ r+1 ] = [Y ~ r+1 , Y [Y i

j

i

j

i

j

and then substituting again (2.8) and (2.9) we obtain for all a = 1, . . . , r ~ (a) , Y ~ (a) ] = [Y i j

s X

~ (a) λkij (x(1) , . . . , x(r) , x)Y k

(2.10)

k=1

and ~j ] = ~i , Y [Y

s X

~k λkij (x(1) , . . . , x(r) , x)Y

(2.11)

k=1

P ~k These vector fields act exclusively in their respective copies of M . sk=1 λkij Y Ps (a) ~ is a vector field in M , and k=1 λkij Y is also a vector field in M (a) , the k r+1 a-th component in the cartesian power M . There is an expression in local ~ coordinates for the vector fields Yi . ~i = Y

n X

ξil (x)

l=1

and ~ (a) = Y i

n X

ξil (x(a) )

l=1

thus, s X

~k = λkij (x(1) , . . . , x(r) , x)Y

n s X X k=1 l=1

k=1

∂ , ∂xl

∂ , ∂xl

λkij (x(1) , . . . , x(r) , x)ξkl (x)

∂ ∂xl

is a vector field in M , and for each a, s X k=1

~ (a) λkij (x(1) , . . . , x(r) , x)Y k

=

n s X X k=1 l=1

λkij (x(1) , . . . , x(r) , x)ξkl (x(a) )

∂ (a)

∂xl

30


~i , Y ~j ], is a vector field in M (a) . Therefore, the coefficients of [Y s X

λkij (x(1) , . . . , x(r) , x)ξkl (x);

(2.12)

k=1

depend only on x1 , . . . , xn ; and analogously for each a varying from 1 to r ~ (a) , Y ~ (a) ], the coefficients of [Y i j s X

λkij (x(1) , . . . , x(r) , x)ξkl (x(a) )

(2.13)

k=1

(a)

(a)

depend only on the coordinate functions x1 , . . . , xn . Fix 1 ≤ α ≤ n. Let

us prove that

∂λkij ∂xα

= 0 for all i = 1, . . . , s, j = 1, . . . , s and k = 1, . . . , s.

The same argument is valid for cartesian power

M r+1 .

∂λkij ∂xβ α

, just interchanging the factors of the

The expressions (2.11) do not depend on xα , and then their partial derivative vanish, s X ∂λkij ξkl (x(a) ) = 0. (2.14) ∂xα k=1

We can write these expressions together in matrix form, 

ξ11 (x(1) )  ξ12 (x(1) )   ..  .  ξ1n (x(1) )   ξ (x(2) )  11  ..  .   ..  .

ξ21 (x(1) ) . . . ξ22 (x(1) ) . . . .. .. . . ξ2n (x(1) ) . . . ξ21 (x(2) ) . . . .. . .. . (r) ξ1n (x ) ξ2n (x(r) ) . . .

   ξs1 (x(1) )  ∂λ1  0 ij   ξs2 (x(1) )  ∂x   . α   ...   .    ..   .   ..  .   .  . (1)     ξsn (x )   ..   ..  =  ;   (2) ξs1 (x )   ..   . .    . ..   .   ..  .   ..       ..  ..  ∂λsij . . ∂xα (r) 0 ξsn (x )

for all i = 1, . . . , s and j = 1, . . . , s. Then, the vector

(2.15)

∂λsij ∂λ1ij ∂xα , . . . , ∂xα

is

a solution of a linear system of n · · · r equations. If we prove the matrix of coefficients above is of maximal rank, then from Lie’s inequality (2.7) we know that this system has only a trivial solution and

∂λkij ∂xα

= 0.


31

In order to do that, let us consider the natural projections, π : M (1) × . . . × M (r) × M → M (1) × . . . × M (r) , onto the first r factors and π1 : M (1) × . . . × M (r) × M → M, onto the last factor. ~ r+1 are projectable by π, and their projection is Vector fields Y k ~ r+1 ) = Y ~ (1) + . . . + Y ~ (r) , π∗ (Y k k k ~ r+1 ) in local coordinates is precisely, The expression of π∗ (Y i ~ r+1 ) = π∗ (Y k

r n X X

ξkl (x(m) )

l=1 m=1

∂ (m)

∂xl

the k-th column of the matrix of coefficients of (2.15). We can state that the ~1 ), . . . , π∗ (Y ~s ) partial derivatives vanish if and only if the vector fields π∗ (Y are generically linearly independents. Assume that there is a non-trivial linear combination equal to zero, s X

~ r+1 ) = Fk (x(1) , . . . x(r) )π∗ (Y k

s X r X

(m)

Fk (x(1) , . . . x(r) )Yk

=0

k=1 m=1

k=1

~ r+1 are generically linearly independent, and then the vector field the fields Y i ~ = Z

s X

~r+1= Fk (x(1) , . . . x(r) )Y k

k=1

s X

~k Fk (x(1) , . . . x(r) )Y

k=1

~ annihilates simultaneously the is different from zero. The vector field Z ∗ sheaves Ψ and π1 (OM ). It annihilates Ψ because it is linear combination of ~ r+1 ; and it annihilates π ∗ (OM ), because it is linear combination of the the Y 1 k ~k . This is in contradiction with the transversality condition of Definition Y ~nr+1 ), . . . π∗ (Y ~nr+1 ) are linearly indepen2.2.5. We have proven that the π∗ (Y ∂λk

dent, and then that the partial derivatives ∂xijα vanish. If we fix a superindex (a) between 1 and r, the same argument is valid for the partial derivatives ∂λkij (a)

∂xα

. Hence, the λkij are constants ckij ∈ C, ~i , Y ~j ] = ckij Y ~k , [Y

(2.16)

~1 ,. . .,Y ~s span a s-dimensional Lie algebra in M , the and the vector fields Y ~ Lie-Vessiot-Guldberg algebra of X.

32


Finite Dimensional Lie-Vessiot-Guldberg Algebra Implies Local Superposition ~ has a finite dimensional Lie-VessiotReciprocally, let us assume that X Guldberg algebra. Throughout this section we assume that the Lie-VessiotGuldberg algebra acts transitively in M . It means that for a generic point ~ at x span the whole tangent space: x of M the values of elements of g(X) ~ x = Tx M. g(X) This assumption of transitivity can be done in the local case without any ~ is not transitive, it gives a non-trivial foliation lose of generality. If the g(X) of M . We can then substitute the integral leaves of the foliation for the phase space M . ~1 , . . . , Y ~m vector fields in M , C-linearly indeLemma 2.3 Let us consider Y pendent but generating a distribution which is generically of rank less than m. ~mr ~ r, , Y Then there is a natural number r such that the lifted vector fields Y 1 generate a generically regular distribution of rank m of vector fields of M r . Proof. We use an induction argument on m. The initial case is true, because a non null vector field generate a distribution which is generically regular of rank 1. Let us assume that the lemma is proven for the case of m vector ~1 , . . . , Y ~m , Y ~ . We can substitute some fields. Consider m + 1 vector fields Y r ~1 , . . . , Y ~m , cartesian power M for M and apply the induction hypothesis on Y so that they generate a generically regular distribution of rank m. ~1 , . . . , Y ~m , Y ~ is not of rank m + 1, then Y can If the distribution spanned by Y be written as linear combination of the others with coefficients functions in M, m X ~i . ~ fi (x)Y Y = i=1

Consider M 2 = M (1) × M (2) . Let us prove that the distribution generated ~ 2 is generically regular of rank m + 1. Using reductio ad ~ 2, . . . , Y ~ 2, Y by Y m 1 absurdum let us assume that the rank is less that m + 1. Then Y 2 is linear ~ 2 with coefficients functions in M 2 , ~ 2, . . . , Y combination of Y 1

m

~2 = Y

m X i=1

~ 2, gi (x(1) , x(2) )Y i

(2.17)


33

on the other hand: ~2 =Y ~ (1) + Y ~ (2) = Y

m X

(1)

~ fi (x(1) )Y i

(2)

~ + fi (x(2) )Y i

(2.18)

i=1

Equating (2.17) and (2.18) we obtain, gi (x(1) , x(2) ) = fi (x(1) ) = fi (x(2) ),

i = 1, . . . , m.

Hence, the functions gi are constants ci ∈ C but in such case, ~ = Y

m X

~i , ci Y

k=1

~ is C-linear combination of Y ~1 , . . . Y ~m , in contradiction with the hypothesis Y of the lemma of C-linear independence of these vector fields. Then, the rank ~2 ~ 2 ,. . .,Y ~ 2 ,Y 2 of the distribution spanned by Y m m is m + 1. 1 ~1 , . . . Y ~s } of g(X). ~ The previous lemma says that there Let us take a basis {Y ~sr is generically ~ r, . . . , Y exist r such that the distribution generated by Y 1 r r regular of rank s in M , the dimension of M is nr so that we have again s ≤ nr. We take one additional factor in the cartesian power, and hence we consider the complex analytic manifold M r+1 . We define Ψ as the sheaf ~ r+1 , . . . , Y r+1 in M r+1 . These os first integrals of the lifted vector fields Y s 1 vector fields span a generically Frobenius integrable distribution of rank s, and then Ψ is a sheaf of regular rings of dimension n + nr − s, which is greater than n. Let us see that the foliation FΨ is generically transversal to the fibers of the projection π1 : M r+1 → M on the last factor. By the transitivity hypothesis ~ the tangent space to these fibers is spanned by the s · r vector fields on g(X), (k) ~ , where i varies from 1 to s and k varies from 1 to r. Let Z ~ be a vector Y i field tangent to the fibers of π1 . If this vector field is tangent to FΨ then it ~ r+1 with coefficients in OM r+1 , is a linear combination of the Y i s X

~ = Z

(1)

Fi (x

(r)

,...,x

~ r+1 = , x)Y i

i=1 k=1

i=1

and from that we obtain that, s X

r s X X

(k) Fi Yi

+

s X

Fi Yi ,

i=1

~i = 0, Fi (x(1) , . . . , x(r) , x)Y

(2.19)

i=1

s X i=1

(1)

Fi (x

(r)

,...,x

~ r+1 = , x)Y i

s X i=1

~r Fi (x(1) , . . . , x(r) , x)Y i

(2.20)

34


~r =Y ~ (1) + . . . + Y ~ (r) . The vector fields Y ~ r ,. . .,Y ~ r span a distribution where Y s 1 i i i which is generically of rank s by construction. And, if considered as vector fields in M r+1 , they do not depend of the last factor in the cartesian power. Then we can specialize the functions Fi to some fixed value x ∈ M in the last component, for which the linear combination (2.19) is not trivial. We obtain an expression, s X

~i = 0, Gi (x(1) , . . . , x(r) )Y

(2.21)

i=1

~i with coefficients functions that gives us is a linear combination of the Y r ~ if M . Vector fields Yi are C-linearly independent; therefore this linear combination (2.21) is trivial and the functions Gi vanish. These functions are the restriction of the functions Fi to arbitrary values in the last factor of M r+1 , then the functions Fi also vanish. We conclude that the vector field ~ is zero, and FΨ intersect transversally the fibers of the projection π1 . Z

2.2.2

Proof of Global Lie’s Superposition Theorem

Lie-Vessiot Systems Admit Superposition Laws Consider a faithful pretransitive Lie group action, G × M → M,

(σ, x) 7→ σ(x),

~ is a Lie-Vessiot system relative to the action of G. There and assume that X exists an analytic open subset W ⊂ M r under the hypothesis of Definition 2.2.2: it is a union of principal orbits, and there exist the quotient W/G. In such case the bundle, π : W → W/G, is a principal bundle modeled over the group G. Consider a section s of π defined in some analytic open subset V ⊂ W/G. Let U be the preimage of V for π. The section s allow us to define a function, g : U → G,

x ¯ → g(¯ x) g(¯ x) · s(π(¯ x)) = x ¯

which is nothing but the usual trivialization function. Define, ϕ : U × M → M, ψ : U × M → M,

(¯ x, y) 7→ g(¯ x) · y (¯ x, y) 7→ g(¯ x)−1 · y.


35

~ and Ψ = ψ ∗ OM is Lemma 2.4 The section ϕ is a superposition law for X, ~ a local superposition law for X. Proof. Let us consider the application, ~ 7→ A ~M , A

R(G) → X(M ),

that send a right-invariant vector field in G to its corresponding fundamental ~i vector field in M . Let {A1 , . . . , As } be a basis of R(G), and denote by X M ~ their corresponding fundamental vector fields Ai . Then, ~ =∂+ X

s X

~ i, fi (t)X

i=1

fi (t) ∈ OS .

Let us consider the following vector field in G × S, ~=∂+ A

s X

~i, fi (t)A

i=1

that we call automorphic system. From the definition of fundamental vector ~ then σ(t) · x0 is a solution of fields, we know that if σ(t) is a solution of A, ~ X for any x0 ∈ M .

Now let us consider the map g : U → G. By construction it is an isomorphism of G-spaces; it verifies g(σ · x ¯) = σ · g(¯ x). Therefore, it maps fundamental vector fields in U to fundamental vector fields in G. Let us remember that fundamental vector fields in G are right invariant vector fields. We have that ~ r is projectable by Id × g, X Id × g : S × U → S × G,

~ r 7→ A, ~ X

~ r is the automorphic vector field A. ~ and the image of X ~ x(1) (t), . . . , x(r) (t). We Finally, let us consider r particular solutions of X, denote by x ¯(t) the curve (x(1) (t), . . . , x(r) (t)) in M r . Therefore, x ¯(t) is a ~ r in M r . Hence, g(¯ solution of X x(t)) is a solution of the automorphic system ~ in G, and for x0 ∈ M wi obtain by composition g(¯ A x(t)) · x0 , which is a ~ solution of X. By the uniqueness of local solutions we know that when x0 varies in M we obtain the general solution. Then, ϕ(x(1) (t), . . . , x(r) (t), x0 ) = g(¯ x(t)) · x0 ~ is the general solution and ϕ is a superposition law for X.

36


With respect to ψ, let us note that it is a partial inverse for ϕ in the sense, ψ(¯ x, ϕ(¯ x, x)) = x,

ϕ(¯ x, ψ(¯ x, x)) = x.

~ r+1 then ϕ(¯ Hence, if (¯ x(t), x(t)) is a solution of X x(t), ψ(¯ x(t), x(t)) = x(t) ~ is a solution of X and then ψ(¯ x(t), x(t)) is a constant point x0 of M , and ~ r. Ψ = ψ ∗ OM is a sheaf of first integrals of X 2

Superposition Law Implies Lie-Vessiot Relative to Pretransitive Lie Group Action ~ Consider a superposition law for X, ϕ : U × M → M. First, let us make some consideration on the open subset U ⊂ M r in the definition domain of ϕ. Consider the family {(ϕλ , Uλ )}λ∈Λ of different su~ ϕλ defined in Uλ × M . There is a natural perposition laws admitted by X, partial order in this family: we write λ < γ if Uλ ⊂ Uγ and ϕλ is obtained from ϕγ by restriction: ϕγ |Uλ ×M = ϕλ . For a totally ordered subset Γ ⊂ Λ we construct a supreme element by setting, [ Uγ , UΓ = γ∈Γ

and defining ϕΓ as the unique function defined in UΓ compatible with the restrictions. By Zorn lemma, we can assure that there exist a superposition law ϕ defined in some U × M which is maximal with respect to this order. From now on we assume that the considered superposition law is maximal. It is clear that ϕ¯ : U × M r → M r ,

(¯ x, y¯) 7→ (ϕ(¯ x, y (1) ), ϕ(¯ x, y (2) ), . . . , ϕ(¯ x, y (r) )),

~ r in M r . For is a superposition formula for the lifted Lie-Vessiot system X each x ¯ ∈ U we consider the map σx¯ : M → M,

x 7→ ϕ(¯ x, x).

It is clear that σx¯ is a complex analytic automorphism of M . We denote by σ ¯x¯ for the cartesian power of σx¯ acting component by component, σ ¯x¯ : M r → M r ,

y¯ 7→ ϕ(¯ ¯ x, y¯).

The map σ ¯x¯ is a complex analytic automorphism of M r .


37

Lemma 2.5 (Extension Lemma) If x ¯ and y¯ are in U and there exist z¯ ∈ M r such that ϕ(¯ ¯ x, z¯) = y¯, then z¯ is also in U . Proof. Let us consider x ¯, y¯, z¯ in U such that ϕ(¯ ¯ x, z¯) = y¯. Then we have a commutative diagram MO B BB BBσx¯ BB B σy¯ ! /M M

σz¯

and it is clear that σz¯ = σy¯σx−1 ¯ , or equivalently ϕ(¯ z , ξ) = ϕ(¯ y , σx−1 ¯ (ξ))

(2.22)

for all ξ in M . Now let us consider x ¯ and y¯ in U and z¯ ∈ M r verifying the same relation ϕ(¯ ¯ x, z¯) = y¯. We can define ϕ(¯ z , ξ) as in equation (2.22). Let V be the set of all r-frames z¯ satisfying a relation such as stated above. Let us see that V is an open subset containing U . Relation ϕ(¯ ¯ x, z¯) = y¯ is equivalent to z¯ = σ ¯x−1 (¯ y ), and then it is clear that ¯ [ V = σ ¯x−1 ¯ (U ), x ¯∈U

which is union of open subsets, and then it is an open subset. The superposition law ϕ naturally extends to V × M , by means of formula (2.22). Because of the maximality of this superposition law, we conclude that U coincides with V and then such a z¯ is in U . 2 Lemma 2.6 Denote G the following set of automorphisms of M , G = {σx¯ | x¯ ∈ U }. G is a group of automorphisms of M . Proof. Let us consider two elements σ1 , σ2 of G. They are respectively of ~ r . From the form σx¯ , σy¯ with x ¯ and y¯ in U . ϕ¯ is a superposition law for X the local existence of solutions for differential equations we know that there exist z¯ ∈ M r such that ϕ(¯ ¯ x, z¯) = y¯. From the previous lemma we know that −1 z¯ ∈ U . We have that σy¯σx−1 ¯ = σz¯. Then σ2 σ1 is in G and it is a subgroup of the group of automorphisms of M . 2 Now let us see that G is endowed with an structure of complex analytic Lie group; and also that the Lie algebra R(G, M ) of its fundamental fields in M

38


~ of X. ~ The main idea is to contains the Lie-Vessiot-Guldberg algebra g(X) translate infinitesimal deformations in U to infinitesimal deformations in G. This approach goes back to Vessiot [Ve1893.b]. Consider x ¯ in U , and a tangent vector ~vx ∈ Tx¯ U . We can project this vector to M by means of the superposition principle ϕ(¯ x, x). Letting x as a free ~ variable we define a vector field V in M , ~ϕ(¯x,x) = ϕ′ V vx ); (¯ x,x) (~ where ϕ′(¯x,x) is the tangent map to ϕ at (¯ x, x): ϕ′(¯x,x) : T(¯x,x) (U × M ) → Tϕ(¯x,x) M. This assignation is linear, and then we obtain a map Tx¯ U → X(M ),

~ ~vx 7→ V

that identifies Tx¯ U with a finite dimensional space of vector fields in M which we denote by gx¯ . Let us see that this space does not depends on x ¯ in U . Consider another r-frame y¯ ∈ U . There exist a unique z¯ ∈ U such that ϕ(¯ ¯ x, z¯) = y¯. Consider the map, Rz¯ : U → M r ,

ξ 7→ ϕ(ξ, ¯ z¯),

Rz¯(¯ x).

~ induced in M by We have ϕ(¯ ¯ y , ξ) = ϕ(R ¯ z¯x ¯, ξ), and then the vector field V the tangent vector ~vx ∈ Tx¯ U is the same vector field that the induced by the tangent vector Rz′¯(~v ) ∈ Ty¯U . Then we conclude that gx¯ ⊆ gy¯. We can gave the same argument for the reciprocal by taking w ¯ such that ϕ(¯ ¯ y , w) ¯ = x ¯. Then gx¯ = gy¯. We denote by g this finite dimensional space of vector fields in M and s its complex dimension. This space g is a quotient of Tx¯ U , so that Lie’s inequality s ≤ nr holds. Let us see that g is a Lie algebra. We can invert, at least locally, the superposition law with respect to the last component, x = ϕ(¯ x, λ),

λ = ψ(¯ x, x).

Denote by Ψ the sheaf generated by the components ψi of these local inver~ the induced sions; Ψ is a local superposition law. Consider ~v ∈ Tx¯ U and V r ~ ~ vector field V ∈ g. It is just an observation that Vx¯ , the value at x ¯ of the


39

~ also induces the same vector field V ~ . Then, in the r-th cartesian power of V language of small displacements , ~x = ϕ(¯ ~x¯r , λ), x + εV x + εV

~x¯r , x + εV ~x ). λ = ψ(¯ x + εV

~ r+1 ψi = 0, and Ψ is a sheaf of first integrals of r + 1 cartesian powers Then V of the fields of g. Then we can follow the argument of subsection 2.2.1 and conclude that g spans a finite dimensional Lie algebra. Now let us consider the surjective map π : U → G. For σ ∈ G the stalk π −1 (σ) is defined by analytic equations, π −1 (σ) = {¯ x ∈ U | ∀x ∈ M ϕi (¯ x, x) = σ(x)}.

(2.23)

Let us see that this stalk is a closed sub-manifold of U . Let us compute the tangent space to π −1 (σ) at x ¯. A tangent vector ~v ∈ Tx¯ U is tangent to the fiber of σ if and only if ~v is into the kernel of the canonical map Tx¯ U → g. Therefore, the fiber of ~x has constant dimension, so that it is defined by a finite subset of the equations (2.23). Hence, the stalk π −1 (σ) is a closed submanifold of U of dimension nr − s. In such case there is a unique analytic structure on G such that π : U → G is a fiber bundle. Consider gr the Lie algebra spanned by cartesian power vector fields, ~r =V ~ (1) + . . . + V ~ (r) , V ~ ∈ g. This is a Lie algebra canonically isomorphic to g. The fields with V of gr are projectable by π. The Lie algebra g is then identified with a Lie algebra of vector fields in G, the Lie algebra R(G) of right invariant vector fields. We deduce that g is the algebra R(G, M ) of fundamental fields of G in M .

Finally, let us see that the elements of R(G, M ) span the Lie-Vessiot-Guldberg ~ For all t0 ∈ S we have, algebra of X. ~ tr )x¯ , λ) = x + (X ~ t )x , ϕ(¯ x + ε(X 0 0

~ t is the vector field of g induced by the tangent vector (X ~ r )x¯ and then X t0 0 ~ t ∈ g for all t ∈ S and therefore there are V ~i ∈ g and at any x ¯ in U . Then X analytic functions fi (t) ∈ OS for i = 1, . . . , s such that, ~ =∂+ X

s X

~i , fi (t)V

i=1

~ is a Lie-Vessiot system in M related to the action of G. and X

40


Lemma 2.7 The action of G on M is pretransitive. Proof. First, by Lemma 2.5 the open subset U is union of principal orbits. Let us prove that the space of orbits U/G in a complex analytic manifold. Consider π : U → G, x ¯ 7→ σx¯ as in the previous lemma. Let U0 be preimage of Id, which is a closed submanifold of U . Consider the map, π2 : U → U,

x ¯ 7→ σx−1 ¯, ¯ x

then π(π2 (¯ x)) = Id, and the image of π2 is U0 . The two projections π : U → G and π2 : U → U0 give a decomposition U = U0 × G and then the quotient U/G ≃ U0 is a complex analytic manifold. We conclude that the action of G on M is pretransitive. 2 ~ is a Lie-Vessiot system associated to the pretransitive We conclude that X action of G on M .

2.3

Automorphic Systems

The automorphic system is the translation of a Lie-Vessiot system to the Lie group G. This approach is due to Vessiot. From now on, consider the a complex analytic connected Lie group G, and a faithful pre-transitive action of G on M . ~ in M with coefficients in the Riemann Consider a Lie-Vessiot system X surface S. Then, s X ~ i, ~ fi (t)X X =∂+ i=1

~ i are fundamental vector fields in M . Consider the where the vector fields X natural map, ~ 7→ A ~M , R(G) → X(M ), A ~ i to applying right invariant vector fields to fundamental fields. Let us call A M ~ =X ~ i. the element of R(G) such that A i ~ to the Definition 2.3.1 We call automorphic vector field associated to X non-autonomous vector field in G, ~=∂+ A

s X i=1

~i. fi (t)A

2.3. Automorphic Systems

41

Reciprocally let N be an homogeneous G-space; we call Lie-Vessiot system ~ to the non-autonomous vector field, induced in N by A ~N = ∂ + A

s X

~N fi (t)A i .

i=1

~ is the Lie-Vessiot system A ~ M induced in M by A. Let us note that X

2.3.1

Solution Space

Superposition Law for the Automorphic System The action of G on itself is transitive. Right invariant vector fields in G are ~ is a particular fundamental fields in G. Then, the automorphic system A ~ case of a Lie-Vessiot system. Hence, there is a superposition law for A. ~ At t0 ∈ S, the tangent vector σ ′ (t0 ) Consider σ(t) a local solution of A. ~ is (At0 )σ(t0 ) . Consider any τ ∈ G; let us define a new curve γ(t) in G as the composition σ(t) · τ . The tangent vector to the curve γ(t) at t0 is ~ t )σ(t ) ) = (A ~ t )γ(t ) . Hence, γ(t) is another γ ′ (t0 ) = Rτ′ (σ ′ (t0 )) = Rτ′ ((A 0 0 0 0 ~ solution of A. Proposition 2.3.1 The composition law G × G → G in G is the superpo~ sition principle for A. Proof. Consider a solution σ(t). As stated above, for all τ in G, σ(t) · τ is ~ Let us see that this is the general solution. Consider another solution of A. t0 in S. For each τ in G, σ(t) · τ is the solution curve of initial conditions t0 7→ σ(t0 )·τ . The action of G on itself -by the right side- is free an transitive; and all initial conditions are obtained in this way. 2 ~ Then σ(t)·τ (t)−1 Corollary 2.8 Consider σ(t) and τ (t) two solutions of A. is a constant point of G.

Structure of Solution Space The particularity of Lie-Vessiot systems is that certain finite sets of solutions give us the general solution. For the automorphic systems, this structure is even simpler. Any particular solution gives us the general solution. There

42


is no difference between the notion of general and particular solution. This property implies that the solution space for this system has a simple structure. ~ be an automorphic system in G depending on S. Proposition 2.3.2 Let A ′ Consider S ⊂ S such that there exist an analytic solution σ : S ′ → G. Then ~ defined in S ′ is a principal homogeneous space the space of solutions of A with an action of G by the right side. ~ the space of solutions of A ~ defined in S. The superProof. Consider Sol(A) ~ by the right side, position law gives us have an action of G on Sol(A) ~ × G → Sol(A), ~ Sol(A)

(σ(t), τ ) → Rτ (σ(t)).

This action is free and transitive, by uniqueness of solutions. The space of ~ is a principal homogeneous space. solutions Sol(A) 2

Hierarchy of Lie-Vessiot Systems Let us consider the following objects: two G-spaces M and N , an auto~ in G depending on the Riemann surface S, and the morphic vector field A ~ M and A ~ N in M and N respectively. induced Lie-Vessiot systems A Let f be a surjective morphism of G-spaces, f : M → N,

f (σ · x) = σ · f (x).

The map f applies fundamental vector fields of the action of G on M to fundamental vector fields of the action of G on N . Thus, f transforms ~ into Lie-Vessiot systems in N . It is clear that it Lie-Vessiot systems in M ~ M into A ~N : transforms A ~M ) = A ~N . f∗ (A ~ N gives us, by composition, a solution curve f (x(t)) A solution curve x(t) of A M ~ of A . We have a surjective map: ~ M ) → Sol(A ~ N ). Sol(A ~ M admits a superposition law ϕ, which is true if the Let us assume that A action of G in M is pretransitive, ϕ : U × M → M,

U ⊂ M r.


43

~ M in M for expressing the general We can also use this superposition law of A N ~ . The composition φ1 = f ◦ ϕ express the general solution of solution of A N ~ ~M . A as function of r particular solutions of A φ1 : U × M → N,

U ⊂ Mr

consider x ¯(t), and x, y ∈ M such that f (x) = f (y). Then φ1 (¯ x(t), x) = φ1 (¯ x(t), y). Then φ1 factorizes and gives un a map, φ : U × N → N,

U ⊂ Mr

(2.24)

~ N in function of r particular solutions of that gives the general solution of A M ~ A . It is not a superposition law, but a different object known as a representation formula. A general theory of representation formulae of solutions of differential equations can be done under this point of view. In particular, the associative property of the action a : G × M → M means that it is a morphism of G spaces. We consider the non autonomous vector ~ ⊗ 1 in G × M ; It is clear that a sends the Lie-Vessiot system A ~⊗1 field A M ~ . to the Lie-Vessiot system A If M is a G-homogeneous space, then there is a surjective morphisms of G-spaces, f : G → M, σ 7→ σ · x0 . In this particular case, the representation formula (2.24) gives us that a ~ gives us the general solution of A ~ M thorough the particular solution of A ~ then σ(t) · x0 is the general action of G on M . If σ(t) is a solution of A, ~ solution of X when x0 moves in M . In particular, if we take σ(t) such that σ(t0 ) = Id, then σ(t) · x0 is the solution of the problem of initial conditions x(t0 ) = x0 . Example.[Linear systems] Let us consider the system, n

dxi X aij (t)xj . = dt j=1

As a vector field it is written ~ = X

X i,j

~ ij , aij (t)X

~ ij = xj ∂ . X ∂xi

44


these vector fields span the lie algebra gl(n, C) of the action of general linear group GL(n, C) on Cn . Let us take coordinates uij in GL(n, C). Direct computation gives us the right invariant vector fields, ~ ij = A

n X k=1

then, ~= A

X

ujk

∂ ∂uik

~ ij . aij (t)A

i,j

The automorphic system is written, X duij = aik (t)ukj , dt k

or in matrix form, d U = A(t)U, dt

U = (uij ), A(t) = (aij (t)).

The solutions of the automorphic system are the fundamental matrices of solutions of the linear system. If U (t) is one such of these matrices, then for each x0 ∈ Cn , x(t) = U (t).x0 is a solution of the linear system. Moreover if we take U (t) such that U (t0 ) = Id the previous formula gives us the global solution with initial conditions x(t0 ) = x0 . Example.[Riccati equations] Let us consider the general Riccati equation, dx = a(t) + b(t)x + c(t)x2 . dt As a vector field it is written: ~ = a(t)X ~ 1 + b(t)X ~ 2 + c(t)X ~ 3, X being,

~2 = x ∂ , X ~ 3 = x2 ∂ . ~1 = ∂ , X X ∂x ∂x ∂x The Lie algebra spanned by these vector fields is the infinitesimal generator of the group P GL(1, C) of projective transformations of the projective line P(1, C), u11 x + u12 , x 7→ u21 x + u22


45

which is a Lie group of dimension 3. In order to make the computation easier, let us consider the following: modulo a finite group of order 2, the group P GL(1, C) is identified with SL(2, C) the group of 2 × 2 matrices with determinant 1. By this isogeny the automorphic system is transformed into the linear system, b d u11 u12 u11 u12 a 2 (2.25) = u21 u22 −c − 2b dt u21 u22 each solution (uij (t)) induces the global solution of the Riccati equation, x(t) =

2.3.2

u11 (t)x0 + u12 (t) . u21 (t)x0 + u22 (t)

Logarithmic Derivative

For each open subset S ′ of the Riemann surface s we denote by O(S ′ , G) the space of analytic maps from S ′ to G; the elements of this space are complex analytic curves in G. For a curve σ(t) ∈ O(S, G) and a point t0 in S ′ we denote by σ ′ (t0 ) to its tangent vector at t0 , which is the image of ∂t0 by the tangent morphism σt′ 0 : Tt0 S → Tσ(t0 ) G. As usually, we identify the Lie algebra R(G) with the tangent space at the identity element TId G. There is only an element of R(G) whose value at σ(t0 ) is σ ′ (t0 ). The value of this right invariant vector field at Id is Rσ′ −1 (t0 ) (σ ′ (t0 )). In such way we can assign to σ a map from S ′ to R(G) that assigns to each t0 ∈ S ′ the right invariant vector field whose value at σ(t0 ) is σ ′ (t0 ). By the identification of R(G) with TId G this map sends t to ′ ′ Rσ(t) −1 (σ (t)). This is precisely Kolchin’s logarithmic derivative. Definition 2.3.2 Let σ ∈ O(S ′ , G) be a curve in G. We call logarithmic derivative l∂(σ(t)) of σ(t) to the map from S ′ to R(G) that assigns to each t0 ∈ S ′ the right invariant vector field whose value at σ(t0 ) is σ ′ (t0 ). The logarithmic derivative is a map, l∂ : O(S ′ , G) → R(G) ⊗C O(S ′ ),

′ ′ σ(t) → l∂(σ(t)) = Rσ(t) −1 (σ (t)).

46


Because of the construction of the logarithmic derivative the following result becomes self-evident. ~ be an automorphic system in G depending on S. Proposition 2.3.3 Let A ′ ~ if and only if, Then τ ∈ O(S , G) is a solution of A ~t l∂(τ ) = A

(2.26)

Expression (2.26) is know as the automorphic equation of the automorphic ~ Solving the automorphic vector field A ~ is equivalent to finding a system A. particular solution for the automorphic equation. Let us recall that the adjoint automorphism Adjσ is the tangent map at the identity element of the internal automorphism of G, G → G,

ξ 7→ σ · ξ · σ −1 .

Logarithmic derivative satisfies the following property with respect to composition. Proposition 2.3.4 (Gauge change formula) Consider σ(t) and τ (t) in O(S, G). The composition σ(t)τ (t) is also an element of O(S, G). We have: l∂(σ(t)τ (t)) = l∂(σ(t)) + Adjσ(t) (l∂(τ (t))). Proof. By direct computation, l∂(σ(t)τ (t)) = Rτ′ (t)−1 σ(t)−1 ((σ(t)τ (t))′ ) = ′ ′ ′ ′ ′ ′ + L′σ(t) (τ ′ (t))) = Rσ(t) Rσ(t) −1 (σ (t))+ −1 Rτ (t)−1 (Rτ (t) (σ (t)) ′ ′ ′ ′ 2 Rσ(t) −1 Lσ(t) Rτ (t)−1 (τ (t)) = l∂(σ(t)) + Adjσ(t) (l∂(τ (t))). Corollary 2.9 For σ(t) ∈ O(S, G), l∂(σ(t)−1 ) = −Adjσ(t)−1 (l∂(σ(t))). Proof. Apply the gauge change formula to the composition σ(t)·σ(t)−1 = Id. 2 Example.[Logarithmic derivative of matrices] We take the canonical basis ∂ ∂uij of the tangent space of GL(n, C). Let us consider U ∈ GL(n, C); the tangent space TU GL(n, C) is identified with the space of n × n complex square matrices. We identify the Lie algebra gl(n, C) with the tangent space at I, the identity matrix. Let U (t) be a time-dependent non-degenerate

2.4. Lie’s Reduction Method

47

matrix. Then U ′ (t) ∈ TU (t) GL(n, C) is the matrix whose coefficients are the derivatives of coefficients of U . In order to identify it with an element of gl(n, C) we have to apply a right transformation, ′ RU −1 (t) : TU (t) GL(n, C) → TId GL(n, C) = gl(n, C).

As RU −1 is a linear map on the functions uij , then it is its own differential; (t)) ′ ′ ′ −1 (t), and then = RU therefore d log(U −1 (t) U (t) = U (t)U dt d log(U ) = U ′ U −1 . dt For a time-dependent matrix A, that we consider as a curve in the Lie algebra gl(n, C), we set the automorphic equation, d log U = A. dt It is equivalent to U ′ U −1 = A,

U ′ = AU,

the linear system defined by A.

2.4

Lie’s Reduction Method

Sophus Lie developed a method of reduction of Lie-Vessiot systems when a particular solution is known. In despite of its lack of popularity, this method is the hearth underlying of most known methods of reduction of differential ordinary equations, as the classical reduction of Riccati equation (see [Da1894] vol I. ch. I-IV), symplectic reduction (see [Br1991] lec. 7), representation formulas of solution of matrix differential equation (see [Sh-Wi1984]), generalized Wei-Norman method (see [Ca-Ra2002]). Another interesting application is the D’Alambert reduction of variational equations to normal variational equations in dynamical systems (see [Mo1999]). It is also equivalent to the method of reduction shown by Cari˜ nena and Ramos (cf. [CGR2001]). Here this method is presented by means of gauge transformations and the automorphic equation. A context in which –in author’s opinion– it is clearly understood. From now on let us consider the following objects: a G-homogeneous space ~ in G depending on the Riemann surface S, an M , an automorphic system A ~ origin point x0 ∈ M . Denote by H the isotropy subgroup of x0 , and by X M ~ ~ the induced Lie-Vessiot system A in M by A.

48

2.4.1


Gauge Transformations

We fiber the extended phase space S × G over the Riemann surface S. It is a trivial principal fiber bundle π : S × G → S. We perform the same operation for M ; we consider then π : S × M → M as an associated bundle of fiber M (see [No1956]). A map σ(t) ∈ O(S, G) is considered as a section of π. This section induces an automorphism Lσ(t) of the principal bundle, Lσ(t) : S × G → S × G,

(t, τ ) 7→ (t, σ(t) · τ ),

and an automorphism of the associated bundle that we denote by the same symbol, Lσ(t) : S × M → S × M, (t, x) 7→ (t, σ(t) · x). Definition 2.4.1 The above automorphisms are called gauge transformations induced by σ(t). This is nothing but Cartan’s notion of repère mobile on the bundle. These are the natural transformations for Lie-Vessiot systems. Gauge transformations are easily understand by terms of the logarithmic derivative. Theorem 2.10 Lσ transforms automorphic systems onto automorphic vector systems, and Lie-Vessiot systems onto Lie-Vessiot system. A map τ (t) is a solution of the automorphic equation (2.26) if and only if Lσ(t) (τ (t)) = τ (t) · σ(t) is a solution verifies, ~ t ) + l∂(σ(t)). l∂(τ (t) · σ(t)) = Adjσ(t) (A Proof. Assume that τ (t) is a solution of the equation (2.26). Then by the ~ t )+ fundamental property of logarithmic derivative, l∂(σ(t)τ (t)) = Adjσ(t) (A l∂(σ(t)). The “if and only if ” condition is attained by considering the inverse gauge transform Lσ(t)−1 . It proves that Lσ(t) maps the automorphic system ~ to the automorphic system B ~ defined by, A ~ t = Adjσ(t) (A ~ t ) + l∂(σ(t)). B ~ M to B ~M. and then it maps also the Lie-Vessiot system A Example. As it is well known, if we consider a linear system x′ = Ax

2


49

and a change of variable z = Bx, being B a time dependent invertible matrix, then z ′ = B ′ x + Bx′ = B ′ B −1 z + BAx = (B ′ + BA)B −1 z, and z satisfies the transformed linear system, z ′ = (B ′ + BA)B −1 z where (B ′ + BA)B −1 = B ′ B −1 + AdjB (A), as above.

2.4.2

Lie’s Reduction Method

Let us recall that we consider x0 a point of the G-homogeneous space M as origin, and we denote by H the isotropy subgroup of x0 . We also denote by H 0 to the connected component of the identity of H. From the canonical inclusion of Lie algebras R(H 0 ) ⊂ R(G) we know that ~ in H 0 is, in particular, an automorphic system an automorphic system B ~ in H 0 naturally extends to a in G. The non-autonomous vector field B non-autonomous vector field in G by right translations. In order to solve the extended non-autonomous vector field it is enough to find a particular ~ in H 0 . Reciprocally, an automorphic system in G restricts to an solution of B automorphic system in H 0 if and only if its Lie-Guldberg-Vessiot algebra is contained in R(H 0 ). The Lie’s method of reduction stands on the following key lemma that characterizes which automorphic systems in G are, in fact, automorphic systems in H 0 . ~ M . Then, A ~ is Lemma 2.11 Assume that x0 is a constant solution of A 0 an automorphic system in H right-invariant vector fields Bi ∈ Ps: there exist 0 ~ ~ R(H ) such that: A = ∂ + i=1 fi (t)Bi .

~ defined in some Proof. For each t0 in S we take a local solution σ(t) of A, ′ ~t = neighborhood S of t0 , with initial condition t0 7→ Id. In S ′ we have A l∂(σ(t)). As σ(t)·x0 = x0 , σ(t) is a curve in H and its logarithmic derivative takes values at R(H). The Lie algebra R(H) coincides with the Lie algebra R(H 0 ) of the connected component of the identity. Finally we conclude that ~ t ∈ R(H). Then the Lie-Guldberg-Vessiot algebra of A ~ is for all t0 ∈ S, A 0 included in R(H). 2 Let us examine the general case of reduction. Assume that we know an ~ M . For each x ∈ M we analytic solution x(t) for the Lie-Vessiot system A denote, Hx0 ,x = {σ ∈ G|σ · x0 = x}.

50


The isotropy group H acts in Hx0 ,x free and transitively by the right side, therefore Hx0 ,x is a principal homogeneous H-space. We construct the following sub-bundle of π : S × G → S. We define H ⊂ S × G, and π1 : H → S the restriction of π in such way that for t0 ∈ S the stalk π1−1 (t0 ) is Hx0 ,x(t0 ) . Then π1 is a principal bundle modeled over H. Let us take a section σ(t) of π1 defined in some S ′ ⊂ S. Thus, in S ′ we have that x(t) = σ(t) · x0 . Let us consider the gauge transformation ~ to an automorphic system B, ~ Lσ(t)−1 . It maps the automorphic system A ~ t − l∂(σ)), ~ t = Adjσ(t)−1 (A B ~ But, Lσ(t)−1 (x(t)) = σ −1 (t)σ(t) · x0 = x0 . Lσ(t)−1 (x(t)) is a solution of B. Thus, we are in the hypothesis of the previous lemma. We have proven the following result. Theorem 2.12 (Lie’s reduction method) Assume that there is a solu~ M defined in a neighborhood of t0 . Then there exists a neightion x(t) of A ′ borhood S of t0 and a gauge transformation defined in S ′ × G that maps the ~ to an automorphic system B ~ in H 0 . automorphic system A For performing Lie’s reduction we need a section of a principal bundle. In general this bundle is not trivial, and then there are no global sections. We have to consider two different cases: compact and non-compact Riemann surfaces. For non-compact Riemann surfaces we use the following result due to Grauert (see [Si1990]). Theorem 2.13 (Grauert theorem) Let S be a complex connected noncompact Riemann surface. Let F → S be a locally trivial complex analytic principal bundle with a connected complex Lie group as structure group. Then there is a meromorphic section of F defined in S. For compact Riemann surfaces we use some results of [GAGA] paper of Serre. If G is algebraic, then complex analytic principal bundles modeled over G are algebraic. In the general case there are not meromorphic global section: algebraic bundles are not locally trivial in Zariski topology. They are locally isotrivial;isomorphic to trivial bundles up to a ramified covering. It is convenient to introduce the following class of groups. Definition 2.4.2 A complex analytic Lie group H is called special if all principal bundle modeled over H has a global meromorphic section.


51

Special groups are linear and connected (see [GAGA] théorème 1). fortunately groups that appear in our integrability theory are special groups. Let us cite the following result ([MRS2008] theorem 8). Theorem 2.14 Let Sp(2n, C) be the symplectic group of n degrees of freedom. (i) Sp(2n, C) is special. (ii) Every connected solvable linear algebraic group is special. (iii) Let H G be a normal subgroup. If H and G/H are special then G is special. If the isotropy group H of x0 in M is special or S is open then the bundle π1 : H → S has a global meromorphic section σ. The gauge transformation ~ to an automorphic system in H 0 . We Lσ−1 maps the automorphic system A have proven the following result. Proposition 2.4.1 Assume that there is a meromorphic solution x(t) of ~ M defined in S. Let us take a point t0 ∈ S and denote by x0 the point A x(t0 ). Let H be isotropy subgroup of x0 in M . Assume one of the following additional hypothesis, (a) H is a special group. (b) S is non compact and H is connected. In such case there is a meromorphic gauge transformation in S × G that ~ to an automorphic system in H 0 . reduces A Example. Consider the Riccati equation dx = a(t) + b(t)x + c(t)x2 dt and suppose that we know a particular solution f . Then let 1 f σ(t) = 0 1 so that f = σ(t) · 0

52


using the linear fractional action of SL(2, C) on the projective line P(1, C) of the above example. The isotropy of 0 is the subgroup H0 of matrices of the form: λ 0 H0 = . µ λ−1 By means of the gauge transformation induced by σ −1 , u11 − f u21 u12 − u22 v11 v12 . = u21 u22 v21 v22 we transform the linear system (2.25) into the reduced system, b d v11 v12 v11 v12 + cf 0 2 = v21 v22 −c − 2b − cf dt v21 v22 which is in triangular form, and then integrable by quadratures. The induced gauge transformation in the projective line maps x to z = x − u, and then z verifies the Riccati equation, dz = (b + 2cf )z + cz 2 dt 1 we find the classical transformation of the Riccati if we set w = 1z = x−u equation to inhomogeneous linear equation (cf. [Da1894] Vol I, chapter II).

dw = −c − (2b + cf )w. dt

2.5

Analytic Galois Theory

Here we present a differential Galois theory for automorphic systems in G that depends meromorphically on a Riemann surface S. Our approach is similar to the tannakian presentation of differential Galois theory. The difference is that here we use the category of G-homogeneous spaces instead of constructions by tensor products. In the tannakian approach, the Galois group stabilizes all meromorphic invariant tensors of the differential equation; equivalently in our approach the Galois group stabilizes all meromor~ This presentation is even phic solutions of Lie-Vessiot systems induced by A. more direct than the classical tannakian approach: we work directly in the category of G-homogeneous spaces. The Galois group and its applications

2.5. Analytic Galois Theory

53

appear naturally. In this frame there is no need of some technical points that usually appear in other presentations of tannakian differential Galois theory. We construct our Galois group as the fiber of a geometrically defined object, the Galois bundle.

2.5.1

Galois Bundle

Let us consider the automorphic system in G, X ~=∂+ ~i, A fi (t)A

and assume that the fi (t) are meromorphic functions in S. Let us consider S × = S \ {poles of fi },

the Riemann surface that we obtain from S by removing the poles of the ~ is a complex analytic automorphic system meromorphic functions fi . Then A × ~M , in G depending on S . For each G-homogeneous space M we consider A the induced Lie-Vessiot system. ~ M ) as the set of solutions of A ~ M defined in S × For each M , let us define M0 (A that are meromorphic at S. Let C(G) be the set of conjugacy classes of closed analytic subgroups of G. To each c ∈ C(G) it correspond a homogeneous space M (c) isomorphic to G/H being H any closed analytic subgroup of G whose class of conjugation is c. When c varies along C(G), M (c) varies along the set of different class of isomorphic G-homogeneous spaces. Finally, we ~ define the set of meromorphic solutions associated to A, [ ~ = ~ M (c) ) M(A) M0 (A c∈C(G)

~ consist of all the different meromorphic solutions of all the Thee set M(A) different Lie-Vessiot systems induced in G-homogeneous spaces. ~ at Definition 2.5.1 For t0 ∈ S × we define the analytic Galois group of A ~ as the subgroup of G that stabilizes the values at t0 of all the t0 , Galt0 (A) ~ meromorphic solutions of all the Lie-Vessiot systems induced by to A. \ ~ = Hx(t0 ) . (2.27) Galt0 (A) ~ x(t)∈M(A)

54


~ is an intersection of closed complex analytic The Galois group Galt0 (A) subgroups of G, and then it is a closed analytic subgroup of G. ~ does not depend on t0 ∈ Lemma 2.15 The class of conjugation of Galt0 (A) × ~ S . Moreover Galt0 (A) depends analytically on t0 in S × . Proof. Consider t0 and t1 in S × . If t0 and t1 are close enough we can ~ defined un a connected neighborhood assume that there is a solution σ(t) of A including t0 and t1 . By a right translation we can assume that σ(t0 ) is the ~ It means that for all meromorphic identity element. Consider τ in Galt0 (A). ~ solution x(t) ∈ M(A), τ · x(t0 ) = x(t0 ). We have x(t) = σ(t) · x(t0 ). Hence, x(t1 ) = σ(t1 )·x(t0 ) and σ(t1 )·τ ·σ(t1 )−1 ·x(t1 ) = σ(t1 )·τ ·σ(t1 )−1 σ(t1 )·x(t0 ) = ~ an σ(t1 ) · τ · x(t0 ) = σ(t1 ) · x(t0 ) = x(t0 ). Then, σ(t1 ) · τ · σ −1 (t1 ) ∈ Galt1 (A) we conclude that, ~ = σ(t1 ) · Galt (A) ~ · σ(t1 )−1 , Galt1 (A) 0 the Galois groups at t0 and t1 are conjugated. It is proven that the conjugacy class of the Galois group it is locally constant; S × is a connected Riemann surface therefore it is constant. Now, let us consider H a subgroup of G of the same class of conjugation ~ The normalizer subgroup that Galt0 (A). N (H) = {σ ∈ G| σH = Hσ}, is the bigger intermediate group H ⊂ Z ⊂ G such that H Z. Let M be the quotient G-homogeneous space G/N (H), and let us consider the natural projection π1 : G → M . Points of M parameterize the class of conjugation of H, to x = [σ] it corresponds the group σ · H · σ −1 , and that the natural action of G on the quotient is nothing but the action of G by conjugation on the conjugacy class of H. This parametrization allow us to define a map h : S × → M that sends t 7→ h(t). The image h(t0 ) is the ~ in its conjugacy class. point of N corresponding to the subgroup Galt0 (A) ~ Near t0 take any solution σ(t) of A such that σ(t0 ) is the identity. Then, ~ = σ(t) · Galt (A) ~ · σ(t)−1 or equivalently h(t) = σ · h(t0 ). Thus, h(t) Galt (A) 0 ~ M induced in M , so that it is an is a solution for the Lie-Vessiot system A ′ analytic function in S . 2 ~ at t0 and M the quotient Lemma 2.16 Consider H the Galois group of A M ~ space G/H. Then A has a meromorphic solution in M .


55

~ is Proof. Consider x0 = [H] the origin point of M . The group Galt0 (A) the isotropy group of the values of all meromorphic solutions of induced LieVessiot systems, as stated in formula (2.27). The complex analytic group G ~ as subgroup of G are: is of finite dimension. The equations of Galt0 (A) ~ = {σ ∈ G | σ(x(t0 )) = x(t0 ) , ∀x(t) ∈ M(A)}. ~ Galt0 (A) ~ gives us some of the equations of Galt (A). ~ As a complex Each x(t) ∈ M(A) 0 ~ analytic manifold G is of finite dimension, thus Galt0 (A) is defined by a finite number of equations, at least locally. Then it suffices to consider a finite number of such meromorphic solutions, y1 (t),. . .,ym (t) each one defined in a ~ ⊂ Hy (t ) . homogeneous space yk : S × → Mk . For each Mk , we have Galt0 (A) k 0 By fixing yk (t0 ) as the origin point, the homogeneous space Mk is identified with the quotient G/Hyk (t0 ) . We have a natural projection of G-spaces pk : M → Mk that maps the origin x0 of M onto yk (t0 ). By considering the cartesian power of those projections, we construct π1 : M → M1 × . . . × Mm , which is an injective morphism of G-spaces, that identifies M with an orbit in the cartesian product. The image of the origin point is (y1 (t0 ), . . . , ym (t0 )). Finally, the meromorphic solution of the Lie-Vessiot system in the cartesian power, (y1 (t), . . . , yn (t)), is contained in the image of π1 , so that it is a ~ M which is meromorphic in S. solution of A 2 ~ depends meromorphically on Corollary 2.17 The Galois group Galt (A) t ∈ S. Proof. Let us recall the proof of the Lemma 2.15. Let us denote by H the analytic differential Galois group in some point t0 ∈ S × . The set of different subgroups of G conjugated with H, id est the class of conjugation of [H], is parameterized by the homogeneous space G/N (H). Let M be the homogeneous space G/H. Let us consider the meromorphic solution x(t) in M of the above lemma given by Lemma 2.16. By construction of x(t) the isotropy Hx(t) is the Galois group at t in S × . H is contained in its normalizer N (H): there is a natural projection of homogeneous G-spaces, M → G/N (Hx(t0 ) ). Let y(t) be the projection of x(t). For t0 near t the group corresponding to ~ with initial y(t) is precisely σ(t) · y(t0 ) where σ(t) is the local solution of A

56


condition for t0 the identity element of G. Then, the isotropy group of y(t) is σ(t) · Hx(t0 ) · σ(t)−1 which is the Galois group at t. This property prolongs to the whole Riemann surface S × and we see that y(t) parameterizes the ~ into its class of conjugacy. group Galt (A) 2 ~ depends meromorphically of S. Thus, we can The Galois group Galt (A) ~ ⊂ S × G, which is meromorphic in S define an analytic sub-bundle Gal(A) × ~ and such that the fiber of t0 ∈ S is precisely Galt0 (A). ~ to the complex analytic in S × Definition 2.5.2 We call Galois bundle of A and meromorphic in S principal bundle, [ π ~ = ~ −−− Gal(A) Galt (A) −→ S × . t∈S ×

2.5.2

Analytic Galois Bundle and Picard-Vessiot Bundle

For this section let us assume that G is GL(E), the group of linear automorphisms of a complex finite dimensional vector space. In this case, the ~ is a system of linear homogeneous differconsidered automorphic system A ential equations with meromorphic coefficients in S. Picard-Vessiot theory is developed for these equations. We can define the algebraic differential ~ following the tannakian formalism (see [RM1990] and Galois group of A [Va-Si2003]). ~ is seen as a meromorphic linear connection ∇E×S The automorphic system A in the vector bundle E × S → S. Let us consider T the category spanned by E trough tensor products, arbitrary direct sums, and their linear subspaces. Denote by T ∇ the category of linear connections spanned by (E × S, ∇E×S ) trough tensor products, arbitrary direct sums, and linear subconnections. The objects of T are linear subspaces of the tensor spaces, ,...,mr (E) = F ⊂ Tnm11,...,n r

r M i=1

E ⊗ni ⊕ (E ∗ )⊗mi .

The objects of T ∇ are linear subconnections of the induced connections in the tensor bundles ,...,mr ,...,mr (E) × S, ∇Tnm1,...,n (V, ∇V ) ⊂ (Tnm11,...,n (E) ). s 1

s

(2.28)


57

As before, we define the Riemann surface S × by removing from S the poles of the coefficients of the differential equations. Each point t in S × defines a fiber functor : ωt : T ∇ → T

that sends vector bundles to their fibers in t. For a subconnection (V, ∇V ) ,...,mr as in (2.28), the fiber on t ∈ S × is a linear subespace Vt of Tnm11,...,n s (E).

It is known that the algebraic differential Galois group is the group of automorphisms of the fiber functor ωt . The representation of this differential Galois group depends on the base point t. In this way we obtain a bundle, that we call the Picard-Vessiot bundle, ~ → S×, P V (A)

~ of automorphisms of ωt . whose fiber in t is the group P Vt (A) ~ into GL(E) We represent the group algebraic differential Galois group P Vt (A) in the following way. Let us consider (V, ∇V ) an object of the category T ∇ . Its fiber in t ∈ S is a vector subspace of certain tensor product ,...,mr Tnm11,...,n (E) = r

r M i=1

E ⊗ni ⊗ (E ∗ )⊗mi .

An element σ ∈ GL(E) induces linear transformations of the tensor product. It is known that σ is in the differential Galois group P Vt (A) if and only if is stabilizes the vector space Vt for all object (V, ∇) of the category T ∇ . ~ = {σ ∈ GL(E) | σ(Vt ) = Vt ∀(V, ∇V ) ∈ Obj(T ∇ )}. P Vt (A)

There is a dictionary between linear connections (V, ∇V ) and meromorphic ~ in algebraic homogeneous solutions of Lie-Vessiot systems in associated to A spaces. Let us consider (V, ∇V ) as above. Let k be the dimension of the fibers of V ; ,...,mr for each t ∈ S × the fiber Vt is a k-plane of Tnm11,...,n r (E). Let us consider the ,...,mr ,...,mr grassmanian variety Gr(k, Tnm11,...,n (E)) of k-planes in Tnm11,...,n r r (E); it is a GL(E)-space. The map, ,...,mr S × → Gr(k, Tnm11,...,n (E)) r

t 7→ Vt ,

~ into the is a meromorphic solution of the Lie-Vessiot system induced by A grassmanian variety. This solution is contained in a GL(E)-orbit, that we denote by M . Hence, the map S × → M,

t 7→ Vt ,

58


~M . is a meromorphic solution of the Lie-Vessiot system A This homogeneous space M is isomorphic to the quotient GL(E)/HVt where HVt is the stabilizer subgroup of the linear subspace Vt ; it is an algebraic subgroup of GL(E). Reciprocally, by Chevalley’s theorem (Theorem B.12), any algebraic group is the stabilizer of certain vector subspace. This means ~ is the group of linear that the algebraic differential Galois group P Vt (A) transformations σ ∈ GL(E) that fix the values in t of all meromorphic solutions of associated Lie-Vessiot systems in algebraic homogenous spaces. We have proven the following. ~ ⊂ P Vt (A). ~ Theorem 2.18 There is a canonical inclusion Galt (A) The analytic Galois group is Zariski dense in the algebraic Galois group. ~ be the set of all the different meromorphic solution of Proof. Let A(A) ~ in algebraic homogeneous all the different Lie-Vessiot systems induced by A ~ ~ GL(E)-spaces. Then, A(A) ⊆ M(A). We have that, \ \ ~ = ~ = Hx(t) . Hx(t) ⊆ P Vt (A) Galt (A) ~ x∈M(A)

~ x∈A(A)

~ is Zariski dense. Let H be the Zariski closure of Let us see that Galt (A) ~ Galt (A). It is an intermediate algebraic subgroup, ~ ⊆ H ⊆ P Vt (A). ~ Galt (A) Let M be G/H. By Lemma 2.16 there is a meromorphic solution of the ~ the algebraic homogeneous space M Lie-Vessiot system in GL(E)/Galt (A); is a quotient of such space. Therefore, there is a meromorphic solution of ~ M and P Vt (A) ~ ⊆ H. A 2 Example. Consider the differential equation, x˙ =

1 . t

It is an automorphic equation in the additive group C. There is an analytic action of C on C∗ , C × C∗ → C∗ (x, y) 7→ x · y = ex y. The associated Lie-Vessiot system is y˙ = y/t. It has meromorphic solutions y = λt. The analytic Galois is contained in the isotropy group 2πiZ ⊂ C; in fact they coincide. However, the algebraic Galois group is the whole additive group.


2.5.3

59

Integration by Quadratures

The Lie’s reduction method, applied to an specific case of homogeneous space, gives us an analytic version of Kolchin theorem on reduction to the Galois group. Theorem 2.19 Assume that the fiber of the Galois bundle ~ → S × , is contained a connected group H ⊂ G. Assume one π : Gal(A) of the additional hypothesis: (1) H is an special group. (2) S is a non-compact Riemann surface. ~ to an Then there is a meromorphic gauge transform of G × S that reduces A automorphic system in H. ~ We can apply an internal Proof. Let M be the homogeneous space Galt0 (A). ~ ⊂ H. There is a natural automorphism of G in order to ensure that Galt0 (A) projection M → G/H. Because of Lemma 2.16, there is a meromorphic ~ M . This solution projects onto a meromorphic solution in G/H. solution of A By Lie’s reduction method, Theorem 2.12, there exist a gauge transformation ~ to R(H). reducing A 2

Quadratures in Abelian Groups If G is a connected abelian group, it is known that the exponential map, R(G) → G,

~ 7→ exp(A), ~ A

is the universal covering of G; in fact it is a group morphism if we consider the Lie algebra R(G) as a vector group. The integration of an automorphic equation in the vector space R(G) is done by a simple quadrature in S; thus the integration of an automorphic equation in G is done by composition of the exponential map with this quadrature: Z t ~ A(τ )dτ , σ(t) = exp t0

where dτ is the meromorphic 1-form in S such that hdτ, ∂i = 1.

60


Solvable Groups ¯ = G/G′ Assume that there is a subgroup G′ G such that the quotient G is an abelian group. We have an exact sequence of groups, ¯ G′ → G → G. ~ on G is projected onto an automorphic vector The automorphic vector field A ~ on G. ¯ G ¯ is a abelian, and then we can find the general solution of B ~ field B by means of the exponential of a quadrature. The quadrature is of the form Z t ~ )dτ, B(τ t0

~ )dτ is a closed 1-form with vectorial values in R(G). This 1-form where B(τ is holomorphic in S × , and meromorphic in S. In the general case this closed 1-form is not exact. We need to consider the universal covering S˜× → S × . S˜× is simply connected, and by Poincare’s lemma every closed 1-form is exact. Then we can define, Z t × ~ ˜ ¯ B(τ )dτ . σ : S → G, t 7→ exp t0

Let t0 be a point of S × and t˜0 a point of S˜× in the fiber of t0 . There is natural action of the Poincare’s fundamental group π1 (S˜× , t˜0 ) on the space of sections O(S˜× , G); this is the monodromy representation. Let Hσ be the isotropy of σ for this action. There is a minimal intermediate covering S × (σ) such that the section σ factorizes. The fiber of the such covering S × (σ) → S × is isomorphic to the quotient π1 (S˜× , t˜0 )/Hσ . = G cGGG {{ GGσ { { GG GG {{ { { / S × (σ) S˜× B BB xx BB xx BB x x B! {xx σ

S×

Some of the ramification points S × (σ) → S × have finite index; we add them to S(σ) obtaining a bigger surface S1 (σ). The projection of S1 (σ) onto S is a ramified covering of certain intermediate surface S × ⊂ S1 ⊂ S. The


61

section σ(t) is meromorphic in S1 (σ). Then, we substitute the Riemann ~ in S1 (σ). We apply surface S1 (σ) for S; σ(t) is a meromorphic solution of B Lie’s reduction method 2.4.1, and reduce our equation to an automorphic equation in G′ with meromorphic coefficients in S1 (σ). We can iterate this process and we obtain then the following theorem: Theorem 2.20 Assume that G is a connected solvable group, and one of the following hypothesis: (a) G is a special group. (b) S is a non-compact Riemann surface. ~ on G is integrable by quadratures of closed Then the automorphic system A meromorphic 1-forms in S and the exponential map in G. Proof. Consider a resolution chain G0 G1 . . . Gn−1 G. We can consider the process above with respect Gn−1 G. If S is non-compact, we are under the hypothesis of Grauert theorem. In the compact case, if G is special, then it is a connected linear solvable group, so Gn−1 is also special. In both cases we can apply Proposition 2.4.1. We reduce the automorphic system to an automorphic system in Gn−1 and take coefficients functions in ~ the corresponding covering of S. We iterate this process until we reduce A to canonical form ∂. 2

2.5.4

Infinitesimal Symmetries

Let us consider the extended phase space S × × G for the automorphic sys~ We are looking for vector field symmetries of the system. It means, tem A. ~ in S × × G such that the Lie bracket verifies [L, ~ A] ~ = λ(t, σ)A. ~ vector fields L This equation defines a sheaf of Lie algebras of infinite dimension of vector fields in S × × G. As stated in [Ath1997], we can differentiate between characteristic and non-characteristic symmetries. Characteristic symmetries ~ The are them that are tangent to the solutions, id est proportional to A. ~ sheaf of Lie algebra of characteristic symmetries is generated by A: for any ~ is a characteristic symmetry of complex analytic function F in S × × G, F A ~ A. Characteristic symmetries form a sheaf of ideals of the sheaf symmetries, and then there is a quotient, the sheaf of non-characteristic symmetries (see [Ath1998], also [Ath1997] for the linear case). On the other hand, we can also consider the sheaf of Lie algebras of transversal symmetries. We say

62


that a vector field in S × × G is transversal if it is tangent to the fibers of the projection onto S. A vector field is transversal if and only if it can be written in the form, ~ = L

s X

~i, Fi (t, σ)A

i=1

Fi ∈ OS × ×G ,

(2.29)

~ i form a basis of the tangent bundle to G. For example, they where the A can form a basis of the Lie algebra R(G) of right invariant vector fields; or alternatively, they can form a basis of the Lie algebra L(G) of left invariant vector fields in G. Both cases lead as to interesting conclusions. ~ be a transversal symmetry of A; ~ direct computation gives that the Lie Let L ~ ~ bracket [L, A] is a transversal vector field. Then, transversal symmetries are defined by the more restrictive equations, ~ dti = 0, hL,

~ A] ~ = 0. [L,

Any symmetry can be reduced to a transversal symmetry by adding a mul~ For any non-transversal symmetry L, ~ we have that L ~ − hdt, Li ~ A ~ tiple of A. is a transversal symmetry. The kernel of such projection is precisely the space of characteristic symmetries. Thus, the sheaf of Lie Algebras of noncharacteristic symmetries is isomorphic with the sheaf of transversal symmetries. For this reason, we restrict our studies to the sheaf of transversal symmetries. ~ the sheaf of Lie algebras of transverDefinition 2.5.3 We denote by T rans(A) ~ sal symmetries of A.

Equation of Transversal Symmetries in Function of Left Invariant Vector Fields ~ be an analytic vector field Let us recall Theorem B.3 in Appendix B. Let L ~ is a left invariant vector field if and only if [L, ~ R] ~ = 0 for all right in G; L ~ invariant vector field R ∈ R(G).

The right invariant vector fields are symmetries of the left invariant vector fields and viceversa. This property lead us to some interesting conclusions. ~ 1, . . . , L ~ s } a basis of L(G), the Lie algebra of left invariant Let us consider {L vector fields in G. An arbitrary transversal vector field in S × G is written


63

in the form ~ = L

s X

~ i. gi (σ, t)L

i=1

Let us set out the equations of transversal symmetries, ~ A] ~ = 0, [L, we expand the Lie bracket, ~ A] ~ =− [L,

s X

~ j (t, σ))L ~ j, (Ag

j=1

and then we obtain the equation for the coefficients gj (t, σ), ~ j (σ, t) = 0, Ag

j = 1, . . . , s.

We have proven the following result, which is an extension of a result for the linear case due to Athorne [Ath1997]. ~ A ~ Then, Theorem 2.21 Consider OS×G the sheaf of first integrals of A. ~ are left invariant vector fields with coefficients transversal symmetries of A ~ A in OS×G , ~ A ~ = L(G) ⊗C OS×G T rans(A) .

Note that the algebra of left invariant vector fields is a finite dimensional ~ Lie algebra of dimension s contained in T rans(A).

Equation of transversal symmetries in function of right invariant vector fields Consider, ~=∂+ A

s X

~i, fi (t)A

i=1

~ = L

s X

~i , gi (t, σ)A

i=1

~1, . . . , A ~ s } is a basis of R(G). Let us denote by ck the constants of where {A ij ~i, A ~ j ] = Ps ck A ~k. structure of the Lie algebra R(G), [A k=1 ij

Let us write the equations for transversal symmetries,    s s k X X X ~ k = 0, ~ i gk (t, σ) − ~ A] ~ = −∂gk (t, σ) − gj (t, σ)ckij  A fi (t) A [L, k=1

i=1

j=1

64


that gives us the s partial differential equations satisfied by the coefficients ~ gk (σ, t) of L,   s s X X ~ i gk − gj ckij  (2.30) fi (t) A ∂gk = − j=1

i=1

Right Invariant Symmetries Consider R(G) as a C-vector space. The group G acts in R(G) through the adjoint representation, G × R(G) → R(G),

(σ, A) 7→ Adjσ (A) = Lσ A,

where Adjσ (A) is the vector field A altered by a left translation of ratio σ. Note that the value of Adjσ (A) at the identity is Rσ′ −1 L′σ (AId ). By the ~ adjoint representation, R(G) is a G-space. Then the automorphic system A ~ in R(G). Let us analyze what is the nature induces a Lie-Vessiot system R ~ ~ defined in S ′ ⊂ S is an analytic of the solutions of R. A local solution of R ′ map S → R(G). We interpret this map as a vector field in S ′ × G tangent to the fibers of the projection onto S ′ . Such a solution is written in form, ~ (t) = V

n X

gi (t)Ai ,

i=1

and the differential equations for the coefficients gi (t) are, ∂gk (t) =

s X

fi (t)gj ckij

(2.31)

i,j=1

which is precisely a particular case of equation (2.30). Then we can state: ~ in R(G) × S are the transversal symmeLemma 2.22 The solutions of R ~ whose restriction to fibers of G × S → S are right-invariant vector tries of A fields. ~ is meromorphic in S complex analytic in Let us consider, as before, that A × ~ ~ meromorphic S . Denote Right(A) the set of transversal symmetries of A in G × S whose restriction to fibers of G × S → S are right invariant vector ~ Right(A) ~ fields. In other words, the space of meromorphic solutions of R. ~ is a finite dimensional Lie subalgebra of the algebra of sections of T rans(A)


65

of dimension less or equal than s. On the other hand, for each t ∈ S we can ~ the space of the values at t of elements of Right(A). ~ We consider Rightt (A), ~ know that Rightt (A) is a Lie subalgebra of R(G), and its class of conjugacy depends meromorphically on t in S. ~ is contained in the cenTheorem 2.23 For all t in S × the group Galt (A) ~ tralizer of Rightt (A). ~ 1 (t), . . . X ~ r (t) a basis of Right(A). ~ Then X ~ i (t) is a set of Proof. Consider X ~ in R(G). For meromorphic solutions of the adjoint equation induced by A × ~ each t ∈ S we have that σ ∈ Galt (A) verifies, ~ ~ Adjσ (X(t)) = X(t), ~ and then σ is in the centralizer of Rightt (A).

2

~ automorphic system in the Corollary 2.24 (Morales-Ramis) Let us A ~ symplectic group Sp(2n, C). If Right(A) contains an abelian algebra of dimension n, then for all t in S × the component of the identity element of the ~ is abelian. analytic Galois group Gal0t (A) Proof. Let us consider the Lie algebra sp(2n, C). It is the Lie algebra of linear Hamiltonian autonomous vector fields in C2n . Consider P the space of homogeneous polynomials of degree 2 in the canonical coordinates in C2n . The space P is a Poisson algebra and Hamilton equations gives us an isomorphism of P with sp(2n, C). For each t in S × we can consider ~ of the Galois group and Rightt (A) ~ as Poisson both the Lie algebra galt (A) ~ subalgebras of P . Theorem 2.23 implies that Galt (A) is contained in the ~ and it implies that the Lie algebra of the Galois centralizer of Rightt (A), ~ commutes with Rightt (A). ~ In terms of Poisson brackets: group galt (A) ~ Rightt (A)} ~ = 0. {galt (A), ~ and Rightt (A) ~ are orthogonal Poisson subalgebras of P . If we Thus, galt (A) ~ assume that Rightt (A) is an abelian subalgebra of P of maximal dimension n then, by [Mo-Ra2001.a], the orthogonal of an abelian subalgebra of maximal ~ is abelian, and the connected dimension is also abelian. Hence galt0 (A) ~ is an abelian group. 2 component Galt0 (A) Remark 2.5.1 The adjoint action of G in R(G) is algebraic. In the case ~ is of the Picard-Vessiot theory, Theorem 2.23 also holds. The group P Vt (A)

66


~ It implies an stronger version of contained in the centralizer of Rightt (A). Corollary 2.24, because the connected component of the algebraic differential ~ is bigger that Gal0t (A). ~ Galois group P Vt0 (A)

3 Differential Algebraic Geometry

This chapter is devoted to differential algebraic geometry. This term is relatively new in mathematical literature. We can state that the differential algebraic geometry is with respect to the differential algebra the same than the classical algebraic geometry is with respect to the commutative algebra. In this sense, the differential algebraic geometry is the study of geometric objects associated with differential rings. It is not clear what are the main objects of differential algebraic geometry. Different authors give us alternative approaches. Here, we are interested in useful tools with the purpose to developing an algebraic Galois theory for automorphic systems. We set the theory of schemes with derivations, which has been developed by Buium [Bu1986], and the theory of differential schemes, which goes back to Keigher [Ke1982, Ke1983], Carra’ Ferro (see [Ca1990]), and achieves its final form in the work of Kovacic [Kov2002]. There are other approaches too, which are not contemplated here.

3.1

Differential Algebra

We present here the preliminaries in differential algebra we need for the further development of differential algebraic geometry. Most results throughout this chapter can be stated in the more general context of partial differential rings and fields. However, we will restrict our discourse to the context of ordinary differential rings, but our theory can be generalized. The main references for this subject are [Ri1950], [Ka1957], [Ko1973]. Some necessary preliminaries on commutative algebra and algebraic geometry for this chapter are given in Appendix A.

68

Chapter 3. Differential Algebraic Geometry

Definition 3.1.1 A differential ring is a pair (A, ∂) of a commutative ring A and a derivation ∂ : A → A. By a derivation we mean an application verifying the Leibnitz rule, ∂(ab) = a · ∂(b) + b · ∂(a). An element a ∈ A is called a constant if it has vanishing derivative ∂a = 0. Whenever it does not lead to confusion, we will write A instead of the pair (A, ∂). Let A be a differential ring. The subset of constants CA , CA = {a ∈ A | ∂a = 0}, is a subring of A. When A is a field we call it a differential field. In such case, the constant ring CA is a subfield of A. Definition 3.1.2 An ideal I ⊂ A is a differential ideal if ∂(I) ⊂ I Note that if I is a differential ideal, then the quotient A/I is also a differential ring. For a subset S ⊂ A we denote [S] for the smallest differential ideal containing S, and {S} for the smallest radical differential ideal containing S. For an ideal I ⊂ A we denote I′ for the smallest differential ideal containing I, namely: X I′ = ∂ i (I). i

Morphisms Definition 3.1.3 A ring morphism ψ : A → B between two differential rings (A, ∂A ) and (B, ∂B ) is called a differential ring morphism if ψ ◦ ∂A = ∂B ◦ ψ. Whenever it does not lead to confusion we denote by the same symbol ∂ the derivations in A and B. If ψ is a differential ring morphism, then for each differential ideal I ⊂ B, the preimage ψ −1 (I) ⊂ A is also a differential ideal. Consider K a differential field. A differential ring A endowed with a morphism K ֒→ A is called a differential K-algebra. If A is a differential field then we say that it is a differential extension of K.

3.1. Differential Algebra

69

Localization Let A be a differential ring, and let S ⊂ A be a multiplicative system. The localized ring S −1 A admits a unique derivation such that the localization morphism is a morphism of differential rings. This derivation is defined as follows: a ∂a · b + a · ∂b ∂ : S −1 A → S −1 A, 7→ . b b2

Keigher Rings and Ritt Algebras Definition 3.1.4 We call Ritt algebra to any differential ring containing the field of rational numbers Q. √ If I ⊂ A is an ideal, we denote I to be its radical ideal, the intersection of all prime ideals containing I. In algebraic geometry, there is a one-toone correspondence between the set of radical ideals of A and the set of Zariski closed subsets of Spec(A), the prime spectrum of A. In order to perform an analogous systematical study of the set of differential ideals - id est differential algebraic geometry - we should require radical of differential ideals to be also differential ideals. This property does not hold in the general case. We have to introduce a suitable class of differential rings. This class was introduced by Keigher (see [Ke1982]); we call them Keigher rings as it is done in [Kov2002]. Definition 3.1.5 A Keigher ring √ is a differential ring verifying that for each differential ideal I, its radical I is also a differential ideal. Definition 3.1.6 For any ideal I ⊂ A we define its differential core as I♯ = {a ∈ I : ∀n(∂ n a ∈ I)}. Keigher rings can be defined in several equivalent ways. The following theorem of characterization includes different possible definitions (see [Kov2002], proposition 2.2.). Theorem 3.1 Let A be a differential ring. The following are equivalent: (a) If p ⊂ A is a prime ideal, then p♯ is a prime differential ideal. (b) If I ⊂ A is a differential ideal, and S is a multiplicative system disjoint from I, then there is a prime maximal differential ideal containing I disjoint with S.

70


(c) If I ⊂ A is a differential ideal, then so is p (d) If S is any subset, then {S} = [S].

√

I.

(e) A is a Keigher ring.

Proposition 3.1.1 If A is a Ritt algebra, then it is a Keigher ring. Proof. See [Ka1957].

2

Proposition 3.1.2 If A is a Keigher ring then for any differential ideal I, A/I is Keigher and for any multiplicative system S, S −1 A is Keigher. Proof. Assume A is Keigher. First, let us prove that A/I is Keigher. Consider thep projection π : A → A/I. Let a be a differential ideal of A/I. Then √ a = π( π −1 (a)) is a differential ideal. Second, consider a localization morphism l : A → S −1 A. Let a ⊂ S −1 A be a differential ideal. Let us denote by b the preimage l−1 (a); it is a differential ideal and l(b) · S −1 A = a. √ √ n Let us consider as ∈ a. as 1s = a1 ∈ a. For certain n, hence a1 ∈ a, an ∈ b √ √ anda ∈ b. A is Keigher, and then ∂a ∈ b. Therefore (∂a)m ∈ b, so that √ ∂a m a. Finally, ∈ a and, for instance, ∂a 1 1 ∈ ∂

a s

=

∂a 1 a ∂s √ − ∈ a, 1 s 1 s2

and by (c) of Theorem 3.1 S −1 A is Kiegher.

2

New Constants From now on let K be a differential field, and let C be its field of constants. We assume that C is algebraically closed. A classical lemma of differential algebra (see [Ko1973] p. 87 Corollary 1) says that if A is a differential Kalgebra, then the ring of constant CA is linearly disjoint over C with K. Let us set this classical lemma in a more geometric frame. Lemma 3.2 Let A be an integral finitely generated differential K-algebra. Then there is an affine subset U ⊂ Spec(A) such that the ring of constants CAU is a finitely generated algebra over C.


71

Proof. Consider Q(A) the field of fractions of A. The extension K ⊂ Q(A) is of finite transcendency degree. Then, K ⊂ K · CQ(A) ⊂ Q(A) are extensions of finite transcendency degree, and there are λ1 , . . . λs in CQ(A) such that K(λ1 , . . . , λs ) = K · CQ(A) . Constants λ1 ,. . .,λs are fractions gfii . Consider the affine open subset obtained by removing from Spec(A) the zeroes of the denominators, s [ U = SpecA \ (gi )0 . i=1

Then, λi ∈ AU and K[CAU ] = K[λ1 , . . . , λs ]. We will prove that CAU = C[λ1 , . . . , λs ]. Let λ ∈ CAU . It is certain polynomial in the variables λi with coefficients in K: X λ= aI λI , aI ∈ K[λi ]M ; I∈Λ

where Λ is a suitable finite set of multi-indices. We can take this set in such way that the {λI }I∈Λ are linearly independent over K, and then so they are over C. {λ, λI }I∈Λ is a subset of K-linearly dependents elements of CAU . By [Ko1973] (p. 87 corollary 1) then they are C-linearly dependent. Hence, λ is C-linear combination of {λI }I∈Λ , λ ∈ C[λ1 , . . . , λs ] and finally CAU = C[λ1 , . . . , λs ]. 2

Tensor Product Let us consider a differential field K and two differential K-algebras A, B. Using the Leibnitz rule we define a derivation in the tensor product A ⊗K B. ∂ : A ⊗K B → A ⊗K B,

a ⊗ b 7→ ∂a ⊗ b + a ⊗ ∂b

This is the only derivation of A ⊗K B such that the canonical injections A → A ⊗K B and B → A ⊗K B are morphisms of differential K-algebras. Hence, it occurs that the tensor product is the direct sum in the category of differential algebras.

almost-constant Differential Rings Definition 3.1.7 Let A be a differential ring. We say that A is almostconstant if every radical differential ideal is generated by constants. The immersion CA → A induce a map, { Radical differential ideals ⊂ A} → { Radical ideals ⊂ CA }.

72


It is self-evident that A is almost-constant if and only if the above correspondence is bijective. almost-constant differential rings have good properties with respect to localization. The following are Propositions 5.2 and 5.3 in [Kov2003]. Proposition 3.1.3 Suppose that A is almost-constant. Let p ⊂ A be a prime differential ideal and q = p ∩ CA . Then, CAp = (CA )q Proposition 3.1.4 Let c ∈ CA . If A is almost-constant, then so is Ac .

3.1.1

Strongly Normal Extensions

Strongly normal extensions were introduced by Kolchin (see [Ko1953]) as an special class of differential field extensions. His aim was to recover the Galois correspondence between subgroups of automorphisms and intermediate extensions in differential Galois theory. Let us remember that in classical Galois theory an algebraic extension k ⊂ K is called normal if k is the fixed field of K by k-algebra automorphisms. For normal extensions there is a one-to-one Galois correspondence between subgroups of automorphisms and intermediate extensions. We say that a differential extension K ⊂ L is weakly normal if K is the fixed field for differential K-algebra automorphisms of L. There is not a Galois correspondence for a weakly normal extension, and then it is necessary to introduce the more restrictive notion of strongly normal extension. Let us consider a differential field K with constant field C of characteristic 0 and algebraically closed. Let us consider an extension of differential fields K ⊂ L. We assume that L is finite differentially generated over K. It means that a finite set of elements of L and its derivatives generate L as Kalgebra. It is proven (see [Ko-La1958]) that a strongly normal extension is, in fact, finitely generated. Thus, we finally deal only with finitely generated extensions. Furthermore, for a strongly normal extension there are no new constants, CK = CL (see [Ko1973] p. 933 proposition 9). The main property of strongly normal extensions is that the group of automorphisms is an algebraic group. Its algebraic structure is shown in [Ko-La1958] by use of Weil’s group chunks. Finally Kolchin avoided this technique by axiomatizing the notion of algebraic group as presented in the


73

monograph [Ko1973], but this last approach is not widespread. A contemporary approach in the frame of scheme theory has been recently done (see [Kov2003], [Kov2006]). We will systematically use this last approach.

Admissible K-Isomorphisms Definition 3.1.8 An admissible K-isomorphism of L is a morphism of differential K-algebras σ : L → L′ , where L ⊂ L′ is some differential extension. Definition 3.1.9 An admissible K-isomorphism σ : L → L′ is strong if (1) σ|CL = IdCL (2) L · σL = L[CL·σL ] One of the main facts about strong isomorphisms is that they are suitable for composition. If σ : L → L′ is strong, then L · σL = L[CL·σL ] and τ : L → L′′ extends to τ : L[CL·σL ] → L′′ [CL·σL ] by the identity in the constant field CL·σL . In this way we can define τ · σ ∈ L → L′′ [CL·σL ]. Definition 3.1.10 We say that K ⊂ L is a strongly normal extension, if every K-isomorphism of L is strong. Let us consider IsoK (L, •) the set of admissible isomorphisms into a certain universal differential extension of L. An universal differential extension of L is a differential extension that includes all differentially finitely generated separable differential extensions of L. This notion is systematically used in Kolchin’s theory. When K ⊂ L is a strongly normal extension, then IsoK (L, •) has a composition law. Kolchin [Ko1973] shows that it is an algebraic group, in the frame of his axiomatic theory . He calls this group the Galois group of the extension. In our scheme approach to algebraic groups, the set of admissible isomorphisms is not exactly an algebraic group, but it can be proven that the functor IsoK (L, •) is the functor of points of the Galois group in Kovacic’s sense: IsoK (L, L′ ) ≃ Gal(L/K)(CL′ ) where Gal(L/K) is the Galois group of the extension, in particular: Aut(L/K) ≃ IsoK (L, L) ≃ Gal(L/K)(C), the set Aut(L/K) of differential K-algebra automorphisms of L is the set of C-points of the scheme Gal(L/K).

74


Definition 3.1.11 Let us consider two admissible K-isomorphisms σ : L → L′ and τ : L → L′′ . Let us consider the differential L-algebras L[σL] ⊂ L′ and L[τ L] ⊂ L′′ . We say that σ specializes to τ , and write σ → τ if there is a commutative triangle of differential L-algebra morphisms, L[σL] .

y< σ yyy yy yy L EE EE E τ EEE "

L[τ L]

We say that σ and τ are related under generic specialization if σ → τ and τ → σ; in such a case we write σ ↔ τ . The relation of specialization is obviously reflexive and transitive. The relation of generic specialization is an equivalence relation. Definition 3.1.12 We call Kolchin topology to the order topology in IsoK (L, •) induced by the specialization relation. The closure of a subset S consist of the admissible isomorphisms to which the elements of S specialize; S¯ = {τ | ∃σ ∈ S σ → τ }. Kolchin topology allows us to set out the Galois correspondence. For the moment, let us enounce the following theorem (see [Ko1953]) with the purpose of going on it with greater detail further. Theorem 3.3 (Galois correspondence) Let K ⊂ L be a strongly normal extension. There is a one-to-one correspondence between intermediate extensions K ⊂ F ⊂ L and Kolchin closed subgroups of IsoK (L, •). To a intermediate extension F, it correspond the subgroup of isomorphisms that induce the identity in F, and to a closed subgroup H it corresponds the field of invariants of H. A closed subgroup is normal if and only if its field of invariants is a strongly normal extension of K.

3.1.2

Lie Extensions

The algebraic differential approach to Lie-Vessiot systems, in terms of differential fields, was initiated by K. Nishioka [Ni1997]. He relates the differ-


75

ential extensions generated by solutions of a Lie-Vessiot system with algebraic dependence on initial conditions; a concept introduced by H. Umemura [Um1985] in relation with Painlevé analysis. He also introduces the notion of Lie extension, a differential field extension that carry the infinitesimal structure of a Lie-Vessiot system. Here we review some of his results, in order to relate them with the Galois theory of automorphic systems. Consider a differential field K of characteristic zero with algebraically closed constant field C. Any considered differential extension of K is a subfield of certain fixed universal extension of K. Definition 3.1.13 We say that a differential extension K ⊂ R depends rationally on arbitrary constants if there exist a differential field extension K ⊂ M such that R and M are free over K and R · M = M · CR·M . It follows that intermediate differential field extensions of a strongly normal extension depend rationally on arbitrary constants. It is expected that, under reasonable conditions, it gives a characterization of those differential field extensions. There are some partial results for the converse. Theorem 3.4 ([Um1985]) Let K be an extension of the complex numbers C generated by a finite number of meromorphic functions in some domain of C. Consider η the general solution of an algebraic differential equation, and K ⊂ R the differential field extension generated by η. If it depends rationally on arbitrary constants, then R is contained in the terminal Km of a finite tower of strongly normal extensions, K ⊂ K1 ⊂ . . . ⊂ Km . Theorem 3.5 ([Ni1989.b]) Let K be algebraically closed and K ⊂ R a differential field extension generated by a single element which is differentially algebraic over K. The following are equivalent: (i) C = CR and R depends on arbitrary constants; (ii) there exists a strongly normal extension of K which contains R. For a differential extension K ⊂ L denote DerK (L) the space of derivations of L that vanish over K. This space is a K-Lie algebra. Definition 3.1.14 We say that a differential extension K ⊂ L is a Lie extension if C = CL , there exists a C-Lie sub algebra g ⊂ DerK (L) such that [∂, g] ⊂ Kg, and Lg = DerK (L).

76


Theorem 3.6 ([Ni1997]) Suppose that K is algebraically closed. Then every intermediate differential field of a strongly normal extension of K is a Lie extension. Theorem 3.7 ([Ni1997]) Let K ⊂ L be a Lie extension and R the maximum between intermediate differential fields depending rationally on arbitrary constants. Then R is also a Lie extension.

3.2 3.2.1

Differential Schemes Differential Spectra

From now on let us consider a differential ring A. Definition 3.2.1 DiffSpec(A) is the set of all prime differential ideals p ⊂ A. Let S ⊂ A any subset. We define the differential locus of zeroes of S, {S}0 ⊂ DiffSpec(A) as the subset of prime differential ideals containing S. This family of subsets define a topology (having these subsets as closed subsets), that we call the Kolchin topology or differential Zariski topology. Note that {S}0 = (S)0 ∩ DiffSpec(A). From that if follows: Proposition 3.2.1 DiffSpec(A) with Kolchin topology is a topological subspace of Spec(A) with Zariski topology. From now on, let us consider the following notation: X = Spec(A), and X ′ = DiffSpec(A). Let us recall that a topological space is said reducible if it is the non-trivial union of two closed subsets. It is said irreducible if it is not reducible. A point of an irreducible topological space is said generic if it is included in each open subset. Proposition 3.2.2 X ′ verifies: (1) X ′ is quasicompact. (2) X ′ is T0 separated.

3.2. Differential Schemes

77

(3) Every closed irreducible subspace of X ′ admits a unique generic point. ′ The map X ′ → 2X , that maps each point x to its Kolchin closure {x} is a bijection between points of X ′ and irreducible closed subspaces of X ′. Proof. see [Ke1983] Proposition 2.1.

2

Here we give some result relating topological properties of X and X ′ . They come from some known results on Keigher rings [Ke1982, Ke1983, Ca1990]. Lemma 3.8 Assume that A is a Keigher ring. Then each minimal prime ideal is a differential ideal. Proof. Then, let p be a minimal prime. By Theorem 3.1 (a), p♯ is a prime differential ideal and p♯ ⊆ p. 2 Proposition 3.2.3 Assume that A is Keigher. Then, X is an irreducible topological space if and only if X ′ is an irreducible topological space. Proof. Just note that the irreducible components of X ′ are the Kolchin closure of minimal prime ideals of A. 2 Proposition 3.2.4 Assume A is Keigher. If X ′ is connected, then X is connected. Proof. Assume that X = Y ⊔ Z, then we have an isomorphism of rings (p1 , p2 ) : A 7→ OX (Y ) × OX (Z),

a 7→ (a|X , a|Y ),

the kernel of each restriction pi is intersection of minimal prime ideals, so by Lemma 3.8 they are differential ideals. Hence, the rings OX (Y ) and OX (Z) are also differential rings. Then, X ′ = Y ′ ⊔ Z ′, being Y ′ = DiffSpec(OX (Y )), Z ′ = DiffSpec(OX (Z)). We have proven that if X disconnects, then X ′ disconnects. 2 Proposition 3.2.5 Assume that A is Keigher. Consider the inclusion i : CA → A and the induced map i∗ : X ′ → Spec(CA ). Then A is almostconstant if and only if i∗ is an homeomorphism.

78


Proof. A is Keigher, and hence there is a one-to-one correspondence between radical differential ideals and Kolchin closed subset of X ′ . If i∗ is an homeomorphism, then each radical differential ideal is generated by constants. 2

Structure Sheaf We define the structure sheaf OX ′ as in [Kov2002]: Let us consider the projection, G π: Ax → X ′ . x∈X ′

F

being x∈X ′ Ax the disjoint union of all the localized rings Ax . We say that a section s of π defined in an open subset U ⊂ X ′ is a regular function if it verifies the following: for all x ∈ U there exist an open neighborhood x ∈ Ux and a, b ∈ A with b(x) 6= 0 (b 6∈ x), such that for all y ∈ Ux with b(y) 6= 0, s(y) = ab ∈ Ay . Thus, a regular function is a section which is locally representable as a quotient. We write OX ′ for the sheaf of regular functions in X ′ . By the above construction we can state: Proposition 3.2.6 The stalk OX ′ ,x is a ring isomorphic to Ax . Theorem 3.9 Let us consider the natural inclusion j : X ′ ֒→ X. Then OX ′ = OX |X ′ . Proof. First, let us define a natural morphism of presheaves of rings on X ′ between the inverse image presheaf j −1 OX and OX ′ . Let us consider an open subset U ⊂ X ′ and a section s of the presheaf j −1 OX defined in U . By definition of inverse image, there is an open subset W of X such that W ∩ X ′ ∩ U and for what s is written as a fraction ab ∈ AW . This fraction is a section of OX ′ (U ), and it defines the presheaf morphism j −1 OX → OX ′ . This presheaf morphism induces a morphism between associated sheaves OX |X ′ → OX ′ . It is clear that this natural morphism induce the identity between fibers (j −1 OX )x = Ax → OX ′ ,x = Ax , and then it is an isomorphism. 2


79

Global Sections Definition 3.2.2 Let Y ⊂ X be any subset. Then, we define the affine stalk of A at Y , AY = lim AV → Y ⊂V

affine

Let us recall that the annihilator ideal ann(a) of an element a ∈ A consist of the elements b of A such that ab = 0. For a ∈ A we denote Ua ⊂ X for the basic open affine subset complementary of (a)0 , the locus of zeroes of a. Definition 3.2.3 An element a ∈ A is a Y -unit if a is contained in the affine open subset Ua ⊂ X. The set of Y -units is denoted U(Y ). An element a ∈ A is a Y -zero if for all x ∈ Y , ann(a) 6⊂ x. The set of Y -zeros is denoted Z(Y ). There is a natural ring morphism j : A → AY . Proposition 3.2.7 U(Y ) is a multiplicative system and Z(Y ) is an ideal. ker(j) = Z(Y ), and AY = U(Y )−1 A. The canonical homomorphism j : A → AY induce a continuous map j ∗ from Spec(AY ) to X. Let us call infinitesimal T affine hull of Y to the intersection b of all affine neighborhoods of Y , Y = Y ⊂V V . V affine

Theorem 3.10 The continuous map j ∗ sends Spec(AY ) to Yb . There is a commutative diagram of continuous maps, ∼ / Yb JJ JJ JJJ j ∗ JJJ $

Spec(AY )

X

Proof. Let us remember that Yb is the intersection of all affine subsets including Y , and an affine subset Ua , with a ∈ A, includes Y if and only if a is a Y -unit. For each a ∈ U(Y ) there are commutative diagrams: / Aa AA AA AA A

AA

AY

UO a X odII II II II II Spec(AY )

80


Let us consider y ∈ Spec(AY ). For each a ∈ U(Y ) the image j ∗ (y) is in Ua , and then j ∗ (y) ∈ Yb . 2

We will denote by Ab the ring of global regular functions OX ′ (X ′ ). Each element a ∈ A define a regular function, so that there is a natural morphism b j : A → A.

Definition 3.2.4 Let a ∈ A. We say that a is a differential unit if for all x ∈ X ′ , a(x) 6= 0. The set of differential units of A is denoted by U(A) or simply U. An element a ∈ A is called a differential zero if is annihilator ann(a) is not contained in any proper prime differential ideal. The set of differential zeros is denoted by Z(A) or simply Z. Proposition 3.2.8 Differential units and zeroes verify: (1) U is a multiplicative system. a ∈ U if and only if a is not contained in any radical differential ideal of A. (2) Z is a differential ideal. √ (3) Z is √ contained in { A}, the smallest radical differential ideal contain√ ing A. In particular if A is Keigher then Z ⊂ A. (4) The kernel of the canonical homomorphism j : A → Ab is Z. (5) The canonical homomorphism j factorizes through U−1 A.

Proof. (1), (2), (4) and (5) are direct consequences of the definitions. (3) is [Kov2002] Proposition 7.4. 2

3.2.2

Differential Schemes

The study of differential schemes started within the work of Keigher [Ke1982, Ke1983] and was continued by Carra’ Ferro [Ca1990], Buium [Bu1986] and Kovacic [Kov2002]. Definitions are slightly different in each author approach, here we follow Kovacic. Definition 3.2.5 A locally differential ringed space is a triple (X, OX , ∂) such that (X, OX ) is a locally ringed space, and ∂ is a derivation of the structure sheaf OX . Whenever it doest not lead to confusion we will write X instead of the triple (X, OX , ∂).


81

Definition 3.2.6 An affine differential scheme is a locally differentially ringed space X which is isomorphic to DiffSpec(A) for some differential ring A. Definition 3.2.7 A differential scheme is a locally differentially ringed space X in which every point has a neighborhood that is an affine differential scheme. Remark 3.2.1 Schemes are differential schemes, endowed with the trivial derivation. The category of differential schemes is an extension of the category of schemes, in the same way that the category of differential rings is an extension of the category of rings. We say that a differential scheme is irreducible, or connected if it is irreducible, or connected, as topological space. We say that it is reduced if its sheaf of regular functions is a sheaf of reduced rings, and we say that it is noetherian if it is quasicompact and the sheaf of regular functions is a sheaf of noetherian rings. By a morphism of differential schemes f : X → Y we mean a morphism of schemes, such that f ♯ : OY → f∗ OX is a morphism of sheaves of differential rings. Let K be a differential field. A K-differential scheme is a differential scheme X provided with a morphism X → DiffSpec(K), it means that OX is a sheaf of differential K-algebras. A morphism of differential schemes f : X → Y between two differential K-schemes is a morphism of differential K-schemes if the sheaf morphism f ♯ : OY → f∗ OX is a morphism of sheaves of differential K-algebras.

Product of Differential Schemes There is not a direct product in the category of differential schemes relative to a given basic differential scheme. This general case is discussed in [Kov1983] and requires the analysis of AAD (annihilators are differential) rings. But in the category of differential K-schemes we can construct direct products by patching tensor products, as it is usually done in classical algebraic geometry. Therefore, DiffSpec(A) ×K DiffSpec(B) = DiffSpec(A ⊗K B).

82


Proposition 3.2.9 Let X and Y be differential K-schemes. The product X ×K Y in the category of differential K-schemes exists. If X and Y are reduced, then so is X ×K Y . Proof. [Kov2003] Proposition 25.2.

3.2.3

2

Split of Differential Schemes

Space of Constants Definition 3.2.8 Let X be a differential scheme. Define the presheaf of rings CX on X by the formula, CX (U ) = COX (U ) . The presheaf CX is a sheaf (see [Kov2003] proposition 26.2). If X is a K differential scheme, then CX is a sheaf of CK -algebras. Proposition 3.2.10 The stalk CX,x in x ∈ X is a ring isomorphic to COX,x and (X, CX ) is a locally ringed space. Definition 3.2.9 We call space of constants of X, Const(X) to the locally ringed space (X, CX ). Definition 3.2.10 We say that X is an almost-constant differential scheme if Const(X) is a scheme. Let X be an almost-constant scheme. Then, each open subset U ⊂ X is also almost-constant. If Y is a reduced closed subscheme of X then Y is almost-constant. In this way if Y is a locally closed reduced subscheme of X, then Y is almost-constant.

Split Let K be a differential field, and C its field of constants. Definition 3.2.11 A differential K-scheme X splits if there is a C-scheme Y and an isomorphism of K-differential schemes, ∼

φ: X − → Y ×C DiffSpec(K). The isomorphism φ is called an splitting isomorphism for X.


83

Proposition 3.2.11 Let X = DiffSpec(A). Suppose that A is reduced. Then A is almost-constant if and only if Const(X) is an affine scheme. In such case Const(X) ≃ Spec(CA ). Proof. [Kov2003] proposition 27.1.

2

Proposition 3.2.12 If X is reduced and split, then it is almost-constant and ∼ X− → Const(X) ×C DiffSpec(K). Proof. [Kov2003] proposition 28.2.

2

Proposition 3.2.13 Suppose that a differential K-algebra A is reduced and almost-constant. Then DiffSpec(A) splits if and only if for all x ∈ DiffSpec(A): Ax = K[CA ]x∩CA Proof. [Kov2003] proposition 28.3.

3.2.4

2

Characterization of Strongly Normal Extensions

Characterization of strongly normal extensions in terms of properties of differential schemes has been recently done by Kovacic [Kov2006], however the main idea goes back to methods of [Bi1962]. From now on let K be a differential field of characteristic 0 with an algebraically closed field of constants C, and K → L a differentially finitely generated differential extension. Theorem 3.11 K → L is a strongly normal extension if and only if the differential scheme DiffSpec(L ⊗K L) splits. In such case denote Gal(L/K) to the scheme Const(DiffSpec(L ⊗K L)). Definition 3.2.12 We say that a strongly normal extension K → L is a Picard-Vessiot extension if Gal(L/K) is an affine scheme. This theorem gives us a parallelism with Galois extensions in classical theory of fields. Note that a field extension k → K is a Galois extension if and only if Spec(K ⊗k K) = G ×k Spec(K) (see [Sa2001]). We also obtain the scheme structure of the Galois group: it is the scheme of constants of DiffSpec(L ⊗K L).

84


Admissible Differential K-Isomorphisms and Galois Group Let IsoK (L, •) be the space of admissible differential K-isomorphisms of L onto a certain universal extension of L. Let σ : L → L′ be and admissible differential K-isomorphism (and then L ⊂ L′ in a canonical way). It induces a morphism of differential L-algebras L ⊗K L → L′ ,

a ⊗ b 7→ σ(a) · b.

There is a natural morphism, ψ : IsoK (L, •) → DiffSpec(L ⊗K L),

σ 7→ xσ = ker(σ ⊗ Id).

From Kovacic’s results we can deduce easily (see [Kov2003] and [Kov2006]): (1) ψ is continuous. (2) The quotient space of admissible differential K-isomorphisms modulo generic specialization, IsoK (L, •)/ ∼, is homeomorphic to the differential spectrum DiffSpec(L ⊗K L). (3) There are morphisms of differential K-schemes (As defined in Appendix B, Section B.2). µ : DiffSpec(L ⊗K L) ×L DiffSpec(L ⊗K L), i : DiffSpec(L ⊗K L) → DiffSpec(L ⊗K L),

such that µ(σ, τ ) = σ −1 · τ and i(σ) = σ −1 . And, if K ⊂ L is strongly normal, then:

(SN1) Each admissible K-isomorphism σ : L → L is an automorphism. (SN2) Aut(L/K) is the set of Kolchin closed points of DiffSpec(L ⊗K L). For a closed differential point σ ∈ DiffSpec(L⊗K L), the quotient field κ(σ) is L. (SN3) The morphisms µ, i are obtained by the base extension C ⊂ K of certain morphisms of C-schemes: µ′ : Gal(L/K)×C Gal(L/K) → Gal(L/K),

i′ : Gal(L/K) → Gal(L/K),

And then, the group Gal(L/K) is an algebraic group over C.

3.3. Schemes with Derivation

85

Galois Correspondence for Strongly Normal Extensions Let us consider as above K ⊂ L a strongly normal extension of differential fields. To each subgroup H ⊂ Gal(L/K) we assign the intermediate extension K ⊂ LH ⊂ L of H-invariants. Reciprocally to each intermediate extension K ⊂ F ⊂ L we assign the subgroup Gal(L/F) ⊂ Gal(L/K) of automorphisms of L that are differential F-algebra automorphism. The Galois correspondence between closed subgroups and intermediate extensions is first shown by Kolchin (see [Ko1953] and [Ko1973]), here we follows the more contemporary presentation of Kovacic. The following result is seen in [Kov2003] (theorems 20.5 and 36.3). Theorem 3.12 The maps H 7→ LH ⊂ L from group subschemes of Gal(L/K) to intermediate differential extensions and F 7→ Gal(L/F) ⊂ Gal(L/K) from intermediate differential extensions subgroup schemes, are bijective and inverse each other. The extension K ⊂ F is strongly normal if and only if Gal(L/F) is a normal subgroup of Gal(L/K). In such case Gal(F/K) is isomorphic to the quotient Gal(L/K)/Gal(L/F).

3.3

Schemes with Derivation

In this section we present some facts of the theory of schemes with derivations. This is mainly the point of view of [Bu1986]. However we consider only regular derivations whereas A. Buium considers the more general case of meromorphic derivations. Our purpose is to relate schemes with derivations to differential schemes. Note that the regularity of the derivation is essential to Theorem 3.13 below; hence it does not hold under Buium’s definition. Let X be a scheme. A derivation ∂X of the structure sheaf OX is a law that assigns to each open subset U ⊂ X a derivation ∂X (U ) of the ring OX (U ). This law is assumed to be compatible with restriction morphisms. Definition 3.3.1 A scheme with derivation is a pair (X, ∂X ) consisting of a scheme X and a derivation ∂X of the structure sheaf OX .

86


Thus, a scheme with derivation is a scheme such that its structure sheaf is a sheaf of differential rings. A morphism of schemes with derivation is a scheme morphism such that induces a morphism of sheaves of differential rings. Let K be a differential field. A K-scheme with derivation is a scheme with derivation (X, ∂) together with a morphism (X, ∂) → (Spec(K), ∂). Thus, the structure sheaf of X is a sheaf of differential K-algebras. Let (X, ∂X ), (Y, ∂Y ) be two K-schemes with derivation. Then the direct product X ×K Y admits the derivation ∂X ⊗ 1 + 1 ⊗ ∂Y . Then, (X ⊗K Y, ∂X ⊗ 1 + 1 ⊗ ∂Y ) is the direct product of (X, ∂X ) and (Y, ∂Y ) in the category of schemes with derivation.

3.3.1

Differential Schemes and Schemes with Derivation

Theorem 3.13 Given a scheme with derivation (X, ∂) there exist a unique topological subspace X ′ ⊂ X verifying (1) X ′ endowed with the structure sheaf OX |X ′ and the derivation ∂|X ′ is a differential scheme. This differential scheme will be denoted Diff(X, ∂). (2) For each open affine subset U ⊂ X, U ∩ X ′ ≃ DiffSpec(OX (U ), ∂). Furthermore, each morphism of schemes with derivation (X, ∂X ) → (Y, ∂Y ) induces a morphism of differential schemes Diff(X, ∂X ) → Diff(Y, ∂Y ). The assignation (X, ∂) ; Diff(X, ∂) is functorial. Proof. If X is an affine scheme then the theorem holds, and X ′ = DiffSpec(OX (X)). Let us consider the non-affine case. Let (X, ∂X ) be an scheme with derivation, and let {Ui }i∈Λ be a covering of X by affine subsets. The ring of sections OX (Ui ) is a differential ring for al i ∈ Λ, and its spectrum Spec(O(Ui )) is canonically isomorphic to Ui .


87

For each i ∈ Λ we take take Ui′ the differential spectrum DiffSpec(OX (Ui )), whichSis a topological subspace of Ui . Then Ui′ ⊂ Ui ⊂ X. Let us define X ′ = i∈Λ Ui′ . Thus, X ′ is a locally differential ringed space with the sheaf OX |X ′ .

Let us prove that X ′ is a differential scheme.

First, let us prove that Ui ∩ X ′ = Ui′ . By construction we have, Ui′ ⊂ Ui ∩ X ′ . Let us consider x ∈ Ui ∩ X ′ . It means that for certain j ∈ Λ, x ∈ Ui ∩ Uj , and x ∈ Uj′ ⊂ Uj . Let us consider an affine neighborhood Ux of x contained in such intersection. Because the inclusion Ux → Uj , we have that x ∈ Ux′ = DiffSpec(OX (Ux )). Then we have inclusions and restriction as follows: / Ui AA AA AA

Ux A

Uj

OX (Ux ) o

OX (Ui )

eLLL LLL LLL L

OX (Uj )

Ux′

/ U′ i ?? ?? ?? ??

Uj′

We conclude that x ∈ Ui′ . Secondly, let us prove that for any affine subset U , the intersection U ∩ X ′ is an affine differential scheme DiffSpec(OX (U )). Let U be an affine subset, and let us denote U ′ the differential spectrum DiffSpec(OX (U )) that we consider as a subset of U . Let us consider x ∈ U ′ . Then, for certain i ∈ Λ, x ∈ U ∩ Ui . Let Ux be an affine neighborhood of x such that Ux ⊂ U ∩ Ui . Denote by Ux′ the differential spectrum of OX (Ux ). We have that Ux′ ⊂ Ui′ , and then x ∈ U ∩ X ′ . Reciprocally let us consider x ∈ U ∩ X ′ . Then for certain i ∈ Λ we have x ∈ Ui′ . By the same argument, we have that x ∈ U is a prime differential ideal of OX (U ). The derivation ∂ induces derivations on the structure sheaf of U ∩ X for each affine open subset U ⊂ X. Then, it induce a derivation ∂ : OX ′ → OX ′ and Diff(X, ∂) = (X ′ , OX |X ′ , ∂|X ′ ) is a differential scheme. Finally, let us consider f : (X, ∂X ) → (Y, ∂Y ) a morphism of schemes with derivation. If we assume that they are both affine schemes, then the theorem holds. In the general case, we cover Y by affine subsets {Ui }i∈Λ , and each fiber f −1 (Ui ) by affine subsets {Vij }i∈Λ,j∈Π . Then f is induced by the family of differential ring morphisms fij♯ : OY (Uj ) → OX (Vij ).

88


These morphisms induce morphisms, fij′ : Vij′ → Ui′ , of locally differential ringed spaces which coincide on the intersections, and then they induce a unique morphism, f ′ : X ′ → Y ′. 2 Definition 3.3.2 Let (X, ∂) be an scheme with derivation. We will say that x ∈ X is a differential point if x ∈ Diff(X, ∂). Corollary 3.14 Let us consider (X, ∂) an scheme with derivation, and x a point of X. Then; the following are equivalent: (a) x ∈ X is a differential point. (b) For each affine neighborhood U , x correspond to a differential ideal of OX (U ). (c) The maximal ideal mx of the local ring OX,x is a differential ideal. (d) The derivation ∂ induces a structure of differential field in quotient field κ(x). (e) The derivation ∂ restricts to the Zariski closure of x.

3.3.2

Split of Schemes with Derivation

Notation. Let Z be a scheme provided with the zero derivation. Then we will write Z instead of the pair (Z, 0). Consider a differential field K and let C be its field of constants. Definition 3.3.3 We say that a K-scheme with derivation (X, ∂) splits, if there is a C-scheme Y , and an isomorphism ∼

φ : (X, ∂) − → Y ×C (Spec(K), ∂), φ is called a splitting isomorphism for (X, ∂).


89

Definition 3.3.4 The space of constants Const(X, ∂) is locally ringed space defined as follows: it is the topological subspace of differential points of X, endowed with restriction of the sheaf of constant regular functions. Proposition 3.3.1 Suppose (X, ∂) is Keigher, then Const(X, ∂) = Const(Diff(X, ∂)). Proof. As topological subspaces of X they coincide by construction. Let X ′ = Diff(X, ∂). If X is Keigher then OX ′ (U ) = lim → OX (V ) (see U ⊆V

[Ca1990]). And because of that we have, ! C

lim OX (V )

→ U ⊆V

= lim COX (V ) , → U ⊆V

and we finish.

2

Definition 3.3.5 (X, ∂) is almost-constant if Const(X, ∂) is a scheme. Proposition 3.3.2 If (X, ∂) splits, then Diff(X, ∂) splits. If (X, ∂) is reduced and split, then it is almost-constant and ∼

(X, ∂) − → Const(X, ∂) ×C (Spec(K), ∂). Proof. Let us consider the splitting isomorphism (X, ∂) → Y ×C (Spec(K), ∂). It is clear that Diff(Y ×C (Spec(K), ∂)) = Y ×C DiffSpec(K). Then the above splitting isomorphism induces the splitting isomorphism of the differential scheme Diff(X, ∂). If X is reduced, then Diff(X, ∂) is also reduced, and then we apply Proposition 3.2.12. 2 Corollary 3.15 Let K ⊂ L finite differentially generated extension. (Spec(L ⊗K L), ∂) splits, then L is strongly normal over K.

If

It is highly expectable that the converse is also true, and then we can characterize strongly normal extensions in term of schemes with derivations. However we do not need the converse in order to develop Galois theory for Lie-Vessiot equations.

4 Galois theory of Algebraic Lie-Vessiot Systems

In this chapter we discuss the Galois theory of Lie-Vessiot systems on algebraic homogeneous spaces. The role of the Riemann surface S of Chapter 2 is correspond here to a differential field K of characteristic zero and with a field of constants C that we assume to be algebraically closed. We study the solutions of algebraic Lie-Vessiot systems. It means that we study the differential extensions of K that allow us to split the Lie-Vessiot system, and the associated automorphic system. We find that they are strongly normal extensions in the sense of Kolchin [Ko1953], and then we can apply Kovacic’s approach to Kolchin’s differential Galois theory. In fact, the Galois theory presented here should be seen as a generalization of the classical Picard-Vessiot theory, obtained by replacing the general linear group by an arbitrary algebraic group. However, as we will see throughout Chapter 5, the particular case of Picard-Vessiot contains all obstructions to solvability, because the non-linear part of an algebraic group over C is an abelian variety: abelian groups do not give obstruction to integration by quadratures.

4.1 4.1.1

Differential Algebraic Geometric theory of Lie-Vessiot Systems Differential Algebraic Dynamical Systems

This thesis begun as a dynamical systems approach to differential algebra. It is a useful task to establish a parallelism between dynamical systems and differential algebraic terminology.

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Chapter 4. Galois theory of Algebraic Lie-Vessiot Systems

From now on let us consider a differential field K, and C its field of constants. We assume that C is algebraically closed and of characteristic zero. Definition 4.1.1 A differential algebraic dynamical system is a K-scheme with derivation (M, ∂M ) such that M is an algebraic variety over K. We say that (M, ∂M ) is non-autonomous if K is a non-constant differential field. There is a huge class of dynamical systems that can be seen as differential algebraic dynamical systems, as polynomial or meromorphic vector fields. It includes Lie-Vessiot systems in algebraic homogeneous spaces, hence it also includes systems of linear differential equations. Furthermore, a differential algebraic study of a dynamical system is suitable in the most general case, but results depend on the choice of an adequate differential field K. For a differential algebraic dynamical system (M, ∂M ) we have the associated differential scheme Diff(M, ∂M ). As a topological space this differential scheme is the set of all irreducible algebraic invariant subsets of the dynamical system. By algebraic, we mean that they are objects defined by algebraic equations with coefficients in K. Let us recall that for a K-algebra L we denote by M (L) the set of L-points of M . This sets consist of all the morphisms of K-schemes from Spec(L) to M , or equivalently, of all the rational points of the extended scheme ML = M ×K SpecL. Definition 4.1.2 Let (M, ∂M ) be a K-scheme with derivation. We call rational solution of (M, ∂M ) any rational differential point x ∈ Diff(M, ∂M ). Let us consider a differential extension K ⊂ L. A solution with coefficients in L is an L-point x ∈ M (L) such that the morphism x : (Spec(L), ∂) → (M, ∂M ), is a morphism of schemes with derivation. In such a case the image x = x(0) is a differential point x ∈ Diff(M, ∂M ) and its quotient field κ(x) is an intermediate extension, K ⊂ κ(x) ⊂ L, we say that κ(x) is the differential field generated by x ∈ M (L).

4.1. Differential Algebraic Geometric theory of Lie-Vessiot Systems

93

As in classical algebraic geometry, there is a one-to-one correspondence between solutions with coefficients in L of (M, ∂M ) and rational solutions of the differential algebraic dynamical system after a base change, (M, ∂M ) ×K (Spec(L), ∂). Solutions of (M, ∂M ) with coefficients in L

change of base field

Rational points of Diff((M, ∂M ) ×K (Spec(L), ∂))

Definition 4.1.3 Let us consider two differential algebraic dynamical systems over K, (M, ∂) and (N, ∂). We say that (M, ∂) reduces to (N, ∂) if there is an algebraic variety Z over C and, (M, ∂) = (N, ∂) ×C Z. The notion of reduction is a generalization of the notion of split. In particular, to split means reduction to (Spec(K), ∂). Given a differential algebraic dynamical system; what does it mean to integrate the dynamical system? As algebraists, we shall use this term for writing down the general solution of the dynamical system by terms of known operations, mainly algebraic operations and quadratures. However, in the general context of dynamical systems there is not a general definition for integrability. We are tempted to say that integrability is equivalent to split. Notwithstanding, there are several situations in which the general solution can be given, but there is not a situation of split. For example, algebraically completely integrable Hamiltonian systems [AMV2002]. In such cases the flux is tangent to a global lagrangian bundle, and the generic fibers of this bundle are affine subsets of abelian varieties. It allows us to write down the global solution by terms of Riemann theta functions and Jacobi’s inversion problem. However, this general solution can not be expressed in terms of the splitting of a scheme with derivation.

94


Split is the differential algebraic equivalent to Lie’s canonical form of a vector field. The scheme with derivation Z ×C (Spec(K), ∂) should be seen as an extended phase space, and ∂ as the derivative with respect to the time parameter. The splitting morphism, (M, ∂) → Z ×C (Spec(K), ∂), can be seen as Lie’s canonical form, usually referred to, in dynamical system argot, as the flux box reduction. Then Z is simultaneously the algebraic variety of initial conditions, and the space of global solutions of the dynamical system. Our conclusion is that the split differential algebraic dynamical systems are characterized by following the property: its space of solutions is parameterized by a scheme. In the context of algebraic Lie-Vessiot systems we will see that algebraic solvability of the problem, is equivalent to the notion of split (Theorem 4.9). And then, this notion plays a fundamental role in our theory. We will see that generically, a Lie-Vessiot equation does not split. If we want to solve it, then we need to admit some new functions by means of a differential extension of K ⊂ L. Thus, the dynamical system splits after a base change to L. The Galois theory will provide us with the techniques for obtaining such extensions and studying their algebraic properties (Proposition 4.2.1).

4.1.2

Algebraic Lie-Vessiot Systems

From now we consider the following objects: – A differential field K of characteristic zero. – The field of constants C of K, that we assume to be algebraically closed. – A connected C-algebraic group G. – A faithful G-homogeneous space M . We denote by R(G) for the Lie algebra of right invariant vector fields in G, and R(G, M ) ⊂ X(M ) for the Lie algebra of fundamental fields. Let us remember that because of the faithfulness of the action of G on M the algebra of fundamental vector fields R(G, M ) is isomorphic to R(G). There is a canonical Lie algebra isomorphism (see Appendix B, Definition B.3.3 and Proposition B.3.1), R(G) → R(G, M ) ⊂ X(M ),

~ 7→ A ~M . A


95

~ in M with Definition 4.1.4 A non-autonomous algebraic vector field X coefficients if K is an element of the vector space X(M ) ⊗C K. ~ in M is written in the form, A non-autonomous algebraic vector field X ~ = X

s X

~ i, fi X

i=1

~ i ∈ X(M ). We define the derivation ∂ ~ for certain elements fi ∈ K and X X ~ as the following derivation of the extended scheme MK : associated to X ∂X~ : OM ⊗ K → OM ⊗ K,

a ⊗ f 7→ ∂a ⊗ f +

s X ~ i f ). (afi ⊗ X i=1

~ in M with Definition 4.1.5 A non-autonomous algebraic vector field X coefficients in K is called a Lie-Vessiot vector field if it is in the vector space R(G, M ) ⊗C K. The differential algebraic dynamical system (MK , ∂X~ ) is called a Lie-Vessiot system in M with coefficients in K. The group G is, in particular, a faithful homogeneous G-space. Let us recall that the Lie algebra of fundamental fields on the group G coincides with the Lie algebra of right invariant vector field R(G). Then, a Lie-Vessiot vector field in G with coefficients in K is an element of R(G) ⊗C K. Definition 4.1.6 We call automorphic vector fields to the Lie-Vessiot vec~ in G with coefficients in K is tor fields in G. An automorphic vector field A an element of R(G) ⊗C K. The canonical isomorphism between R(G) and R(G, M ) allows us to translate Lie-Vessiot vector fields in M to automorphic vector fields in G. Definition 4.1.7 We call automorphic system associated to (M, ∂X~ ) to the ~ is the automorphic vector field whose Lie-Vessiot system (GK , ∂A~ ), where A ~ corresponding Lie-Vessiot vector field in M is X. ~ be a Lie-Vessiot vector field in M , with coefficients in From now on let X ~ K, and let A be the associated automorphic vector field in G.

96

4.1.3


Algebraicity of Superposition Laws

In Chapter 2 we have discussed how to construct a superposition law from a pretransitive Lie group action. The superposition law ϕ and its partial inverse ψ are given by Lemma 2.4. This superposition law is analytic in the general case. For a pretransitive algebraic action of an algebraic group in an algebraic variety we expect to obtain an algebraic superposition laws. It is not true in the general case that algebraic superposition laws exist. It is true in the particular case of algebraic automorphic equation, in which the superposition law is the composition law in the group. However, in the general case, we have to add some suitable hypothesis to obtain the algebraicity of certain superposition law. Here we cover the case of algebraic homogeneous spaces. Let r be the rank of M (Definition B.1.10). In the analytic case we have seen that the set B of the basis is an analytic open subset of M . We need this space and the space of orbits B/G. Hypothesis 1. Let us assume that the set B(k) ⊂ M r (C) of basis of M is the set of rational points of a Zariski open subset B ⊂ M r . Let us assume that there exist geometric universal quotient B/G. In such case π : B → B/G is a principal bundle of structure group G. Principal algebraic bundles are not locally trivial in the general case. They are only iso-trivial, locally isomorphic to a trivial bundle up to a finite morphism [GAGA]. Hypothesis 2. B → B/G.

There exist a meromorphic section s of the bundle

In particular, this hypothesis is satisfied if G is and special group (Definition 2.4.2). Let is denote by W the preimage by π of the domain of definition of s. W is an Zariski open subset of M r formed by principal orbits of the action of G. Once we have this section we can define the superposition law ϕ, and its partial inverse ψ analogously as we did in Lemma 2.4: ψ : W ×C M → M,

ϕ : W ×C M → M,

(b, x) 7→ σs (b)−1 · x (b, x) 7→ σs (b) · x

We have proven the following result, that is partially equivalent to Lemma 2.4 in the algebraic context.


97

Proposition 4.1.1 Let M be a G-algebraic homogeneous space verifying hypothesis 1 and 2. Then there is and algebraic superposition law ϕ for nay Lie-Vessiot systems in M spanned by the fundamental fields of the action of G in M .

4.1.4

Algebraic Logarithmic Derivative

Here we generalize to the geometrical differential algebraic frame the notion of logarithmic derivative defined in Chapter 2 (Definition 2.3.2). Along this text we identify systematically the Lie algebra R(G) with the tangent space Te G = DerC (OG,e , C). It is also important to remark that the tangent space is compatible with extensions of the base field in the following way: ∼

R(G) ⊗C K − → Te (GK ) = DerK (OGK ,e , K).

In classical algebraic geometry it is assumed that derivations of Te (GK ) vanish on K. However, automorphic systems are by definition compatible with the derivation ∂ of K. Thus, the restriction of an automorphic vector field ∂A~ to e ∈ GK is not a tangent vector of Te (GK ): it is shifted by ∂. We have identifications of K-vector spaces: R(G) ⊗C K

∼

/ R(G) ⊗C K + ∂

−∂

/∂ =∂+A ~ ~ A

~ A

/ Te (GK )

/A ~e

Let us consider σ ∈ G(K) and the canonical morphism σ ♯ of taking values in σ: σ ♯ : OGK ,σ → K, f 7→ f (σ). Let us remember that there is a canonical form of extension of the derivation ∂ in K to a derivation in GK . We consider the direct product G ×C (Spec(K), ∂) in the category of schemes with derivation. By abuse of notation we denote by ∂ this canonical derivation in GK . By construction we have that (GK , ∂) splits – the identity is the splitting morphism – and Const(GK , ∂) = G. Let us consider the following non-commutative diagram, OGK ,σ

σ♯

/K . ∂

∂

OGK ,σ

σ♯

/K

(4.1)

98


Lemma 4.1 The commutator σ ′ = [∂, σ ♯ ] of the diagram (4.1) is a derivation vanishing on K, and then σ ′ belong to the tangent space Tσ (GK ) (id est, the space of derivations DerK (OGK ,σ , K)). Proof. [∂, σ ♯ ] is the difference between two derivations, and then it is a derivation. Let us consider f ∈ K ⊂ OGK σ , then σ ′ (f ) = ∂f − ∂f = 0. 2 If σ is a geometric point of GK , then Rσ−1 is a automorphism of GK sending σ to e. It induces an isomorphism between the ring of germs OGK ,σ and OGK ,e , and then an isomorphisms between the corresponding spaces of derivations: Tσ (GK )

R′ −1 σ

/ Te (GK ) ≃ R(G) ⊗C K

Definition 4.1.8 Let σ be a geometric point of GK ; we call logarithmic derivative of σ, l∂(σ), to the automorphic vector fiel Rσ′ −1 ([∂, σ ♯ ]). The logarithmic derivative is then a map: l∂ : G(K) → R(G) ⊗C K. Proposition 4.1.2 Properties of logarithmic derivative: (1) Logarithmic derivative is functorial in K; for each differential extension K ⊂ L we have a commutative diagram: G(K)

/ R(G) ⊗C K

/ R(G) ⊗C L

G(L) (2) Let us consider σ and τ in G(K):

l∂(στ ) = l∂(σ) + Adjσ (l∂(τ )) (3) Let us consider σ ∈ G(K): l∂(σ −1 ) = −Adjσ (l∂(σ)). Proof. (1) comes directly from the differential field extension, (2) is exactly as in Proposition 2.3.4, and (3) is corollary to (2). 2


99

Automorphic Equation Theorem 4.2 Let us consider K ⊂ L a differential extension. Then σ ∈ G(L) is a solution of the differential algebraic dynamical system (GK , ∂A~ ) ~ if and only if l∂(σ) = A. ~ be its logarithmic derivative. Proof. Let us consider σ ∈ G(L), and let B The space R(G) ⊗C L is canonically identified with the Lie algebra of right invariant vector fields on the base extended L-algebraic group GL : R(G) ⊗C L = R(GL ). ~ is seen as a derivation By this identification, the automorphic vector field B ~ ~ ~ is a derivation B of the structure sheaf OGL . The germ B(σ) at σ of B ♯ of the ring OGL ,σ . The composition with σ give us the tangent vector ~ σ ∈ Tσ (GL ): B ~ (σ) B

/ OG ,σ OGK ,σ S K SSS EE SSS SSSS EEEEσ♯ SSS EE SSS E" ~σ B S)

K

~ at the identity point is, by definition, l∂(σ). Since B ~ is a The value of B ♯ ′ ~ right invariant vector field we have l∂(σ) = Rσ−1 (Bσ ) = σ ◦ B(σ) ◦ Rσ♯ −1 ~ σ is equal to the commutator [∂, σ ♯ ] of Definition 4.1.8. Then, B ~ (σ) hence B is the defect of the diagram (4.1); therefore the following diagram commutes: OGK ,σ

σ♯

/K .

~ (σ) ∂+B

∂

OGK ,σ

σ♯

/K

~ is determined by the commutator B ~ σ = [∂, σ ♯ ] and then it is Furthermore, B unique right invariant vector field in GL that forces the diagram to commute. Let us note that the commutation of the above diagram holds if and only ~ is the unique right if the kernel mσ of σ ♯ is a differential ideal. Then B invariant vector field in GL such that the maximal ideal mσ is a differential ~ σ is the germ in σ of the ideal. Let us note also that, this derivation ∂ + B automorphic derivation ~ ∂B~ = ∂ + B,

100


~ the logarithmic derivative of σ, is the unique element we conclude that B, 2 of R(G) ⊗C L such that σ is a differential point of (GL , ∂B~ ). ~ for the soBecause of that we can substitute the automorphic system A, called automorphic equation: ~ l∂(x) = A

(4.2)

Solving Lie-Vessiot Systems Definition 4.1.9 Let us consider σ ∈ G(K). We call gauge transformation induced by σ to the left translation Lσ : GK → GK . Lemma 4.3 (GK , ∂A~ ) splits if and only if the automorphic equation (4.2) has at least one solution in G(K). Proof. Assume (GK , ∂A~ ) splits. Let us consider the splitting isomorphism ψ : (GK , ∂A~ ) → Z ×C (Spec(K), ∂). Let x be a C-rational point of Z. Let us denote by xK the corresponding K-point of GK obtained after the extension of the base field. Thus, ψ −1 (xK ) is a solution of (4.2). Reciprocally, let us assume that there exists a solution σ of (4.2) in G(K). Let us consider the gauge transformation: Lσ−1 : GK → GK . It applies σ onto the identity element e ∈ GK . But the logarithmic derivative l∂(e) vanishes, so that Lσ−1 transforms ∂A~ into the canonical derivation ∂. We conclude that Lσ−1 is an splitting isomorphism. 2 Lemma 4.4 Assume that (GK , ∂A~ ) splits. In such case we can choose the splitting isomorphism between the gauge transformations of GK . This gauge transformation induces the split of any associated Lie-Vessiot system (MK , ∂X~ ). Proof. We use the same argument as above. If it splits, s : (GK , ∂A~ ) → G ×C (Spec(K), ∂) = (G, ∂), then the preimage of the identity element s−1 (e) = σ is a solution of the automorphic system. So that the gauge transformation Lσ−1 : σ 7→ e maps solutions of (GK , ∂A~ ) to solutions of (GK , ∂) and it is an splitting isomorphism.


101

For any associated Lie-Vessiot system (MK , ∂X~ ), and any point x0 ∈ M (C) we have that Lσ (x0 ) is a solution of (MK , ∂X~ ). So that Lσ sends solutions of the canonical derivation ∂ to solutions of ∂X~ . Thus, its inverse Lσ−1 is an 2 splitting isomorphism for (MK , ∂X~ ). ~ a non-autonomous Lemma 4.5 Let Z be a C-algebraic variety and (ZK , D) ~ splits then (ZK , D) ~ differential algebraic dynamical system over K. If (ZK , D) ~ is almost-constant and Const(ZK , D) ≃ Z. ~ splits. It implies that there exist an C-scheme Proof. Assume that (ZK , D) Y , such that ZK = Y ×C Spec(K). We have that ZK ≃ YK , and then Z ≃ Y . 2 Lemma 4.6 Let Z be a reduced C-scheme. There is a one-to-one correspondence between closed subschemes of Z and closed subschemes with derivation of (ZK , ∂) = Z ×C (Spec(K), ∂). Proof. First, let us consider the affine case. Assume Z = SpecR for a C-algebra R. The ring of constants CR⊗C K is R itself. It follows that Const(ZK , ∂) = Z. By Proposition 3.2.11, R ⊗C K is an almost-constant ring: each radical differential ideal is generated by constants. Because of that there is an one-to-one correspondence between radical ideals of R and radical differential ideals of K.

In the non-affine case, let us consider Y a closed sub-C-scheme of Z. The canonical immersion (YK , ∂) ⊂ (ZK , ∂) identifies Y with a closed sub-Kscheme with derivation of (ZK , ∂). Reciprocally, let (Y˜ , ∂|Y˜ ) be a closed sub-K-scheme with derivation of (ZK , ∂). Let us consider {Ui }i∈Λ an affine covering of Z. The collection {Vi }i∈Λ with Vi = Ui ×C K is then an affine covering of ZK . Each intersection Y˜i = Y˜ |Vi is an affine closed sub-K-scheme of Vi . We are in the affine case: by the above argument there are closed sub-C-schemes Yi ⊂ Ui such that (Y˜i , ∂|Y˜i ) = Yi ×S C (Spec(K), ∂). This family 2 defines a covering of a closed sub-C-scheme Y = i∈Λ Yi of Z. ~ a non autonomous Lemma 4.7 Let Z be a C-algebraic variety and (ZK , D) algebraic dynamical system over K. Let Y ⊂ Z a locally closed subvariety, ~ is tangent to Y , so that (YK , D| ~ Y ) is a sub-K-scheme and assume that D ~ splits then (YK , D| ~ Y ) splits. with derivation. If (ZK , D) Proof. By substituting Z for certain open subset we can assume that Y is closed. Let us consider the splitting isomorphism, ~ → Z ×C (Spec(K), ∂). ψ : (ZK , D)

102


~ Y ) is a locally closed subscheme with derivation of The image ψ(YK , D| 2 Z ×C (Spec(K), ∂). By Lemma 4.6 it splits. Lemma 4.8 Assume that the action of G on M is faithful. Then (GK , ∂A~ ) splits if and only if (MK , ∂X~ ) splits. Proof. Lemma 4.4 says that if (GK , ∂A~ ) splits, then (MK , ∂X~ ) splits. Reciprocally, let us assume that (MK , ∂X~ ) splits. For each positive number r we consider the natural lifting to the cartesian power (MKr , ∂ r~ ). The splitting X of (MK , ∂X~ ) induces the splitting of those cartesian powers differential algebraic dynamical system (MKr , ∂ r~ ). For r big enough there is a point x ∈ M r X such that its orbit Ox is a principal homogeneous space isomorphic to G (see Appendix, remark B.3.4). Then (Ox,K , ∂X~ ) is a locally closed sub-K-scheme with derivation of (MKr , ∂ r~ ). By Lemma 4.7 it splits. We also know that X 2 (Ox,K , ∂X~ ) is isomorphic to (GK , ∂A~ ). Finally, (GK , ∂A~ ) splits. Theorem 4.9 Assume that the action of G on M is faithful. Then the following are equivalent. (1) The automorphic equation (4.2) has a solution in G(K) (2) (GK , ∂A~ ) splits. ~ to 0. (3) There is a gauge transformation of GK sending A (4) (MK , ∂X~ ) splits. (5) (GK , ∂A~ ) splits, is almost-constant, and Const(GK , ∂A~ ) ≃ G. (6) (MK , ∂A~ ) splits, is almost-constant, and Const(MK , ∂X~ ) ≃ M . Proof. Equivalence between (1) and (2) comes from Lemma 4.3. Equivalence between (2) and (3) comes from Lemma 4.4. (2) and (4) are equivalent by Lemma 4.8. By Lemma 4.5, they all imply (5) and (6). 2

4.2

Splitting Field of an Automorphic System

Note that a differential extension K ⊂ L, induces a canonical inclusion, R(G, M ) ⊗C K ⊂ R(G, M ) ⊗C L;

4.2. Splitting Field of an Automorphic System

103

so that a Lie-Vessiot vector field with coefficients in K is a particular case of a Lie-Vessiot vector field with coefficients in L. So that if (MK , ∂X~ ) is a Lie-Vessiot system, then (ML , ∂X~ ) makes sense. Definition 4.2.1 We say that a differential extension K ⊂ L is a splitting extension for (MK , ∂X~ ) if (ML , ∂X~ ) splits. From theorem 4.9, we know that K ⊂ L is a splitting extension of (MK , ∂X~ ) if and only it is a splitting extension of (GK , ∂A~ ). Then we will center our ~ attention in the automorphic vector field A.

4.2.1

Action of G(C) on GK

For each σ ∈ G(C), Rσ is an automorphism of GK . The composition law is an action of G on GK by the right side, GK ×C G → GK . ~ is right invariant, so that we expect the differential points The vector field A of (GK , ∂A~ ) to be invariant under right translations. In fact, the above morphism is a morphism of schemes with derivation, (GK , ∂A~ ) ×C G → (GK , ∂A~ ). We apply the functor Diff, and then we obtain an action of the C-algebraic group G on the differential scheme Diff(GK , ∂A~ ), Diff(GK , ∂A~ ) ×C G → Diff(GK , ∂A~ ). Assume that (GK , ∂A~ ) split. In such case, when we apply the functor Const to the previous morphism, we obtain a morphism of schemes, Const(GK , ∂A~ ) ×C G → Const(GK , ∂A~ ). Because of the split we already knew that Const(GK , ∂A~ ) is a C-scheme isomorphic to G. Furthermore, the above morphism says that the action of G by the right side on this G-scheme is canonical. We have proven the following: Lemma 4.10 Assume that (GK , ∂A~ ) splits. Then Const(GK , ∂A~ ) is a principal G-homogeneous space by the right side.

104

4.2.2


Existence and Uniqueness of the Splitting Field

Lemma 4.11 There is a differential point x ∈ Diff(GK , ∂A~ ) which is closed in the Kolchin topology. Proof. Let us consider the generic point p0 ∈ GK . In particular it is a differential point p0 ∈ Diff(GK , ∂A~ ). If p0 is Kolchin closed, then we finish and the result holds. If not, then the Kolchin closure of p0 contains a differential point point p1 such that p0 specializes on it p0 → p1 . We continue this process with p1 . As GK is an algebraic variety, and then a noetherian scheme, this process finish in a finite number of steps and lead us to a Kolchin closed point. 2 Lemma 4.12 Let x ∈ Diff(GK , ∂A~ ) be a closed differential point. Then its field of quotients κ(x) is a differential extension of K with the same field of constants; Cκ(x) = C. Proof. Reasoning by reductio ad absurdum let us assume that there exists c ∈ Cκ(x) not in C. Let us consider an affine open neighborhood U of x and denote by A its ring of regular functions. We identify x with a maximal differential ideal x ⊂ A. Denote by B the quotient ring A/x. B is a differential subring of the differential field κ(x). By Lemma 3.2 there exist b ∈ B such that the ring constants CBb – of the localized ring Bb – is a finitely generated C-algebra. By reducing our original neighborhood U – removing the zeros of b – we can assume that b is invertible and then the localized ring Bb is just B. CB is a non-trivial finitely generated C-algebra over C, because it contains an element c not in C. So that there is a non-invertible element c2 ∈ CB . The principal ideal (c2 ) is a non trivial differential ideal in B. Let us consider a regular function a2 such that a2 (x) = c2 . Then ∂A~ a2 ∈ x and (a, x) is a non-trivial differential ideal of A strictly containing x. We arrive to contradiction with the maximality of x. 2 Proposition 4.2.1 Let x ∈ Diff(GK , ∂A~ ) be a closed point. Then K ⊂ κ(x) ~ is a splitting extension of (GK , A). Proof. Let x be a closed point. Then the canonical morphism x♯ of taking values in x, x♯ : OGK ,x → κ(x) is a morphism of differential rings. Let U be an affine neighborhood of the image of π(x) by the canonical projection


105

π : GK → G. By composition we construct a morphism Spec(κ(x)) → U , OG (U ) π♯

OGK ,x

σ♯

/ κ(x) . n7 n n nnn nnn n n nn x♯

The morphism σ ♯ is the dual of a morphisms σ from Spec(κ(σ)) to U . In other words, σ is a point of G(κ(x)). We consider σ as a rational differential point of (Gκ(x) , ∂A~ ), and then it is a solution of the automorphic equation. 2 By Lemma 4.3, (Gκ(x) , ∂A~ ) splits. Definition 4.2.2 We say that σ, as defined in the above proof, is the fun~ associated with the closed differential point x. damental solution of A Let us consider the action of G on GK by right translations. The derivation ∂A~ is invariant by right translations, and then it is a morphism of schemes with derivation: (GK , ∂A~ ) ×C G → (GK , ∂A~ ) We apply the functor Diff, thus we obtain a morphism of differential schemes which is an algebraic action of G on the set of differential points. Diff(GK , ∂A~ ) ×C G → Diff(GK , ∂A~ ) Proposition 4.2.2 The action of G(C) on the set of closed points of Diff(GK , ∂A~ ) is transitive. Proof. Let us consider a Kolchin closed point x ∈ Diff(GK , ∂A~ ). Let L be the rational field of x. It is an splitting field for (GK , ∂A~ ). We have that (GL , ∂A~ ) splits, hence Diff(GL , ∂A~ ) is an almost-constant differential scheme. Thus Diff(GL , ∂A~ ) is homeomorphic to the principal homogeneous G-space Const(GL , ∂A~ ). The differential extension K ⊂ L induces a commutative diagram of schemes with derivation, (GL , ∂A~ ) ×C G

/ (GL , ∂ ~ ) A

π1

(GK , ∂A~ ) ×C G

/ (GK , ∂ ~ ) A

106


and thus, a commutative diagram of differential schemes, Diff(GL , ∂A~ ) ×C G

/ Diff(GL , ∂ ~ ) . A

π2

Diff(GK , ∂A~ ) ×C G

/ Diff(GK , ∂ ~ ) A

Let s be a Kolchin closed point of Diff(GK , ∂A~ ). The projection π2 of the above diagram is exhaustive. Consider any p ∈ π2−1 (s), and let us consider a Kolchin closed point x in the closure {p}. Thus, π2 (x) is in the closure {s}. As s is a Kolchin closed point we know that π2 (x) = s. Hence, there is a Kolchin closed point x ∈ Diff(GL , ∂A~ ) such that π2 (x) = s. Consider two Kolchin closed points s, y ∈ Diff(GK , ∂A~ ). Because of the above argument there are two Kolchin closed points x, y ∈ Diff(GL , ∂A~ ) such that π2 (x) = s and π2 (y) = y. The set of Kolchin closed points of Diff(GL , ∂A~ ) is a G(C)-homogeneous space in the set theoretical sense. Then there is σ ∈ G(C) such that x · σ = y, and by the commutativity of the diagram we have s · σ = y. 2 Corollary 4.13 Let x and y be two closed points of Diff(GK , ∂A~ ). Then there exists an invertible K-isomorphism of differential fields κ(x) ≃ κ(y). Proof. There is a closed point σ ∈ G, such that x · σ = y. Then Rσ : (GK , ∂A~ ) → (GK , ∂A~ ) is an automorphism that maps x to y. Then it induces an invertible K-isomorphism Rσ♯ : κ(y) → κ(x). 2 Definition 4.2.3 For each closed point x ∈ Diff(GK , ∂A~ ) we say that the differential extension K ⊂ κ(x) is a Galois extension associated to the nonautonomous differential algebraic dynamical system (GK , ∂A~ ). Notation. As we have proven, all Galois extensions associated to (GK , ∂A~ ) are isomorphic. From now on let us choose a closed point x and denote by K ⊂ L its corresponding Galois extension.


107

Proposition 4.2.3 A Galois extension is a minimal splitting extension for (GK , ∂A~ ) in the following sense: If K ⊂ S is any splitting extension for (GK , ∂A~ ) then there is a K-isomorphism of differential fields L ֒→ S. Proof. If K ⊂ S is an splitting extension, then (GS , ∂A~ ) splits. Hence, for each Kolchin closed differential point x ∈ Diff(GS , ∂A~ ) the rational field of x is S. Let us consider the natural projection π : (GS , ∂A~ ) → (GK , ∂A~ ). We can choose a Kolchin closed point x ∈ Diff(GK , ∂A~ ) such that π(x) = x. We have a morphism of K-differential algebras between the corresponding rational fields π ♯ : L → S. 2 Example.[Picard-Vessiot extensions] Let us consider system of n linear differential equations ∂x = Ax, A ∈ gl(n, K),

and let us denote aij for the matrix elements of A. The algebraic construction of the Picard-Vessiot extension is done as follows (cf. [Ko1973] and [Va-Si2003]):

Let us consider the algebra K[uij , ∆], being ∆ = |uij |−1 the inverse of the determinant. Note that it is the algebra of regular functions on the affine group GL(n, K). If is an affine group, and then it is isomorphic to the spectrum GL(n, K) = Spec(K[uij , ∆]). We define the following derivation,

∂A~ uij =

n X

aik ujk ,

k=1

that gives to K[uij , ∆] the structure of differential K-algebra, and to (GL(n, K), ∂A~ ) the structure of automorphic system. The set of Kolchin closed differential points od Diff(GL(n, K), ∂A~ ) is the set of maximal differential ideals of R. A Picard-Vessiot algebra is a quotient algebra K ⊂ K[uij , ∆]/m, and a Picard-Vessiot extension is a rational differential field K ⊂ κ(m). It is self-evident that the Picard-Vessiot extension is the particular case of Galois extension when the considered group is the general linear group. Lemma 4.14 Let K ⊂ S be a splitting extension. The canonical projection π : Diff(GS , ∂A~ ) → Diff(GK , ∂A~ ) is a closed map.

108


Proof. It is enough to prove that the projection y = π(y) of a closed point y ∈ Diff(GS , ∂A~ ) is a closed point. Let us take a closed point z ∈ {y}. Then π −1 (z) is closed and there is a closed point z ∈ π −1 (z). Diff(GS , ∂A~ ) is a principal homogeneous G-space, there is a σ ∈ G(C) such that z · σ = y, and then z · σ = y. G(C) acts transitively in the space of closed points, and z is closed, so that we have proven that y is closed. In fact y and z are the same differential point. 2 Proposition 4.2.4 Let us consider any intermediate differential extension, K ⊂ F ⊂ S, with K ⊂ S an splitting extension. The projection, π : Diff(GF , ∂A~ ) → Diff(GK , ∂A~ ), is a closed map. Proof. Let us consider the following diagram of projections: π1

Diff(GS , ∂A~ )

PPP PPPπ2 PPP PPP '

/ Diff(GK , ∂ ~ ) A nn7 πnnnn nnn nnn

Diff(GF , ∂A~ )

By Lemma 4.14 π1 and π2 are closed and surjective. Then π is closed.

2

Lemma 4.15 Let K ⊂ F ⊂ L be an intermediate differential extension of the Galois extension of (GK , ∂A~ ), and σ the fundamental solution associated to x. Let us consider the sequence of base changes, Diff(GL , ∂A~ ) σ

π1

/ Diff(GF , ∂ ~ ) A

/

y

π2

/ Diff(GK , ∂ ~ ) A

/

x

,

then y is closed in Kolchin topology, κ(y) is the Galois extension L and σ is the fundamental solution associated with y. Proof. By Proposition 4.14 π1 is a closed map, so that y is a closed point. The chain of projections induces a chain of differential extensions κ(x) ⊆ κ(y) ⊆ κ(σ) but κ(x) = κ(σ), and then we have the equality. 2


4.2.3

109

Galois Group

Here we give a purely geometrical definition for the Galois group associated to a Kolchin closed differential point. We prove strong normality of the Galois extensions, and identify our geometrically-defined Galois group with the group of automorphisms of the Galois extension. Let us consider the action of G on Diff(GK , ∂A~ ) shown in Subsection 4.2.1: Diff(GK , ∂A~ ) ×C G → Diff(GK , ∂A~ ). Definition 4.2.4 Let x ∈ Diff(GK , ∂A~ ) be a Kolchin closed differential point. We call Galois group of the system (GK , ∂A~ ) in x to the isotropy subgroup of x in G by the above action, and denote it by Galx(GK , ∂A~ ). Proposition 4.2.5 Galx(GK , ∂A~ ) is an algebraic subgroup of G. Proof. Denote by Hx the Galois group in x. Let us consider the projection π1 from GK to G induced by the extension C ⊂ K. Denote by x the point π1 (x), and let U be an affine neighborhood of x. Then U = G \ Y with Y closed in G. UK is an affine neighborhood of x in GK . We have that the ring of regular functions in UK is the tensor product OG (U ) ⊗C K. We identify x with a maximal prime differential ideal x ⊂ OG (U ) ⊗C K. Let us consider a C-point σ of G. Then, for each f ∈ OG (U ) ⊗C K we have that the right translate Rσ♯ (f ) is in OG (U · σ −1 ) ⊗C K. The morphism π2 : G → G,

σ 7→ Rσ (x),

is algebraic, and let W be the complementary in G of π2−1 (Y ), W = G \ π2−1 (Y ), W is an open subset in G verifying: (a) for all σ ∈ W (C), x ∈ U ∩ U · σ −1 , (b) Hx ⊂ W .

110


We will prove that the equations of Hx in W are algebraic. Let us consider W1 an affine open subset in W . Let {ξ1 , . . . , ξr } be a system of generators of OG (W ) as C-algebra. The composition is algebraic, π3 : U ×C W1 → G,

(y, σ) 7→ y · σ,

and it induces a morphism, π3♯ : OG,x → (OG (U ) ⊗C O(W1 ))π−1 (x) , 3

and then for each f ∈ OG,x , π3♯ (f ) = F (ξ), is a rational function in the ξi with coefficients in OG,x . We identify x with a prime ideal of OG (U ) ⊗C K. We consider a system of generators, x = (η1 , . . . , ηr ),

ηi ∈ OG (U ) ⊗C K.

Property (b) says that by the natural inclusion, j : OG (U ) ⊗C K → (OG (U ) ⊗C O(W1 ))π−1 (x) ⊗C K, 3

j(x) spans a non trivial ideal of (OG (U ) ⊗C O(W1 ))π−1 (x) ⊗C K, and then we 3 have a commutative diagram: OG (U ) ⊗C K

/ (OG (U ) ⊗C O(W1 ))π−1 (x) ⊗C K . 3

π4

κ(x)

/ (κ(x) ⊗C O(W1 ))π−1 (x) 3

An element σ ∈ W1 stabilizes x if and only if Rσ♯ (ηi ) ∈ x, and this is so if and only if π4 (j(ηi )) = 0 for i = 1, . . . , r. Let us consider a basis {eλ }λ∈Λ of κ(x) over C. For each i, we have a finite sum: P Giα (ξ)eα , π4 (j(ηi )) = Pα β Hiβ (ξ)eβ and then Giα (ξ) ∈ O(W1 ) are the algebraic equations of Hx in W1 .

2

Remark 4.2.1 Let x be a Kolchin closed differential point as above, and H ⊂ G the Galois group of (GK , ∂A~ ) in x. Then HK = H ×C Spec(K) is the stabilizer subgroup of x by the action of composition by the right side: GK ×K GK → GK .


111

However, the morphisms Rσ for σ ∈ HK are not in general morphisms of schemes with derivation. In the same sense, for any field extension K ⊂ L, HL ⊂ GL is the stabilizer group of π −1 (x) where π is the natural projection from GL to GK . This means that HL stabilizes the fiber, in the following sense: for each L-point σ ∈ HL , Rσ : GK → GK induces, Rσ |π−1 (x) : π −1 (x) → π −1 (x). Proposition 4.2.6 Consider two Kolchin closed differential points x, y in Diff(GK , ∂A~ ). The groups Galx(GK , ∂A~ ) and Galy(GK , ∂A~ ) are isomorphic conjugated algebraic subgroups of G. Proof. The group of C-points of G acts transitively in the set of closed differential points. Hence, there exists σ ∈ G(C) with x · σ = y, and then Hx · σ = σ · Hy. 2 Theorem 4.16 The Galois extensions associated to (GK , ∂A~ ) are strongly normal extensions. Proof. Let us consider a Galois extension K ⊂ L. Thus, L is the rational field of certain Kolchin closed differential point that we denote by x. Let us consider σ ∈ GL the fundamental solution associated to x. We have that σ projects onto x and the gauge transformation Lσ−1 is a splitting morphism. We define the morphism ψ of schemes with derivation trough the following commutative diagram: (GL , ∂A~ ) O

Lσ

π

/ (GK , ∂ ~ ) A h3 h h h hhh h h h hh hhhh ψ hhhh

(GL , ∂) = G ×C (Spec(L), ∂)

Denote by H the Galois group in x. We have that (HL , ∂) ⊂ (GL , ∂) is a closed subscheme with derivation. The group HL is the preimage of H by the projection from GL to G. By remark 4.2.1 HL is the stabilizer of the fiber of x in GL . It means that for any point z of GL that projects to x and any L-point τ of HL , the right translate z · τ also projects onto x. In particular we have that ψ(τ ) = x, and then (HL , ∂) ⊂ ψ −1 (x).

112


Reciprocally, let us consider an L-point τ ∈ ψ −1 (x). Therefore π(σ · τ ) = x. The following diagram is commutative: GL ×L GL

/ GL

/ GK

GK ×K GK

We deduce that, for any other preimage σ ¯ of x by π, the right translated σ ¯ ·τ also projects onto x. Thus, τ stabilizes the fiber of x, so that τ ∈ (HL , ∂). Finally we have the identity: ψ −1 (x) = (HL , ∂) = H ×C (Spec(L), ∂). On the other hand we apply the affine stalk formula (Proposition A.2.3, that comes from the classical stalk formula, Theorem A.2, in Appendix A) to x. We obtain the isomorphism: π −1 (x) ≃ (Spec(L ⊗K L), ∂). From the definition of ψ we know that Lσ gives us an isomorphism between the fibers π −1 (x) and ψ −1 (x). This restricted morphism Lσ |(HL ,∂) is a splitting morphism π / (Spec(L ⊗K L), ∂) kk5 {x} O

Lσ |(HL ,∂)

kkk kkk k k kkkk ψ kkk

H ×C (Spec(L), ∂)

of the tensor product L ⊗K L. By corollary 3.15, K ⊂ L is a strongly normal extension. 2 Remark 4.2.2 Following [Kov2003], DiffSpec(L ⊗K L) is the set of admissible K-isomorphism of L, modulo generic specialization. In the case of a strongly normal extension K ⊂ L the space of constants Const(DiffSpec(L ⊗K L)) is an algebraic group and its closed points correspond to differential K-algebra automorphisms of L. Let us consider the previous splitting morphism, H ×C (Spec(L), ∂) → (Spec(L ⊗K L), ∂) if we apply the constant functor Const, we obtain a isomorphism of Calgebraic varieties, s H− → Gal(L/K),


113

where H and Gal(L/K) are algebraic groups. To each τ ∈ H, we have x · τ = x, and the Rτ♯ : L → L. We have Rτ♯ ◦ Rτ♯¯ = Rτ♯ τ¯ and it realizes H as a group of differential K-algebra automorphisms of L. Theorem 4.17 The Galois group Galx(GK , ∂A~ ) is the group of differential K-algebra automorphisms of the Galois extension K ⊂ κ(x). Proof. Denote, as above, by H ⊂ G the Galois group and by L the Galois extension κ(x). We consider the isomorphism s stated in remark 4.2.2. Let us prove that s is an isomorphism of algebraic groups over C, and that for τ ∈ H(C), s(τ ) is the automorphism Rτ♯ of L, induced by the translation Rτ . We already know that s is a scheme isomorphism. We have to prove that it is a group morphism. For τ ∈ H, let us compute s(τ ). First, let us denote by τ¯ the point of HL obtained from τ after the base extension from C to L. It is a differential point of (HL , ∂). Then Lσ (¯ τ ) = Rτ (σ) ∈ π −1 (x). We identify −1 Rτ (σ) with a differential point of π (x). By the stalk formula we have that π −1 (x) = (Spec(OGK ,x ⊗K L), ∂). We identify Rτ (σ) with a prime differential ideal of OGK ,x ⊗K L. Because π(Rτ (σ)) = x, the morphism Rτ (σ)♯ factorizes, OGK ,x ⊗K L

QQQ QQQRτ (σ)♯ QQQ QQQ QQQ (/ κ(x) ⊗K L L x♯ ⊗Id

ψ

and then the kernel of ψ is the prime differential ideal defining the automorphism s(τ ), ψ(a ⊗ b) = s(τ )(a) · b Let us consider the right translation Rτ , GL

GK

Rτ

/ GL

σ

/ GK

x

Rτ

/ Lσ (¯ τ) /x

114


we have a commutative diagram between the local rings, LO o

Id

LO

Rτ (σ)♯

σ♯

OGL ,π−1 (x) o

R♯τ

O

OGK ,x o

,

OGL ,π−1 (x) O

R♯τ

OGK ,x

where OG,π−1 (x) = OGK ,x⊗K L, and the morphism Rτ♯ on these rings is defined as follows: OGK ,x ⊗K L → OGK ,x ⊗K L,

a ⊗ b 7→ Rτ♯ (a) · b.

It is then clear that morphism ψ defined above sends, ψ : (a ⊗ b) 7→ Rτ♯ (a) · b and then its kernel defines the automorphism Rτ♯ and we finally have found Rτ♯ = s(τ ). 2

4.2.4

Galois Correspondence

There is a Galois correspondence for strongly normal extensions (theorem 3.12). It is naturally transported to the context of algebraic automorphic systems. Let L be a Galois extension, which is the rational field κ(x) of a Kolchin closed point x as above. Let F be an intermediate differential extension, K ⊂ F ⊂ L. We make base extensions sequentially so that we obtain a sequence of schemes with derivations, (GL , ∂A~ ) → (GF , ∂A~ )) → (GK , ∂A~ ), and the associated sequence of differential schemes, Diff(GL , ∂A~ ) → Diff(GF , ∂A~ ) → Diff(GK , ∂A~ ).


115

Let σ ∈ G(L) be the fundamental solution induced by x. We obtain a sequence of differential points: σ 7→ y 7→ x. They are Kolchin closed and σ is the fundamental solution associated to x and y (Lemma 4.15). The stabilizer subgroup of y is a subgroup of the stabilizer subgroup of x. We have inclusions of algebraic groups, Galy(GF , ∂A~ ) ⊂ Galx(GK , ∂A~ ) ⊂ G. In particular we have that K ⊂ F is a strongly normal extension if and only if Galy(GF , ∂A~ ) Galx(GK , ∂A~ ). Proposition 4.2.7 Assume that Galx(GK , ∂A~ ) is the whole group G, and K ⊂ F is a strongly normal extension. Then the quotient group ¯ = G/Galy(GF , ∂ ~ ) G A ~ be the projection of A ~ in R(G) ¯ ⊗C K. Then, there is a unique exists. Let B ¯ closed differential point z ∈ Diff(GK , ∂B~ ), and, ¯ ¯ K , ∂ ~ ) = G. Galz(G B Proof. The quotient realizes itself as the group of automorphisms of the differential K-algebra F. The extension K ⊂ F is strongly normal, and then this group is algebraic by Galois correspondence (Theorem 3.12). The induced morphism ¯K, ∂ ~ ) π : Diff(GK , ∂A~ ) → Diff(G A restricts to the differential points, and it is surjective. The hypothesis ~ = G implies that Diff(GK , ∂ ~ ) consist in the only point {x}, Galx(GK , A) A ¯ K , ∂ ~ ) = {z}. Hence, z is the generic point of GK and the and then Diff(G A Galois group is the total group. 2 Reciprocally let us consider an algebraic subgroup H ⊂ Galx(GK , ∂A~ ). Then H is a subgroup of differential K-algebra automorphisms of L. Let F = LH be its field of invariants. We have again a sequence of non-autonomous algebraic dynamical systems (GL , ∂A~ ) → (GF , ∂A~ ) → (GK , ∂A~ ).

116


Let again σ be the fundamental solution induced by x, we have the sequence of closed differential points, σ 7→ y 7→ x Proposition 4.2.8 Let us consider an intermediate differential field, K ⊂ F ⊂ L, as above, and H = Aut(L/F), then (a) H is the Galois group Galy(GF , ∂A~ ) ⊂ Galx(GK , ∂A~ ). (b) K ⊂ F is strongly normal if and only if H Galx(GK , ∂A~ ). In such case Aut(F/K) = Galx(GK , ∂A~ )/H. Proof. By considering the identification of the Galois group with the group of automorphisms, the result is a direct translation of the Galois correspondence for strongly normal extensions (see [Kov2003] Theorem 20.5, Theorem 3.12 in this text). 2 In particular, each algebraic group admits a unique normal subgroup of finite index, the connected component of the identity. Let Gal0x (GK , ∂A~ ) be the connected component of the identity of Galx(GK , ∂A~ ) and, Gal1x (GK , ∂A~ ) = Galx(GK , ∂A~ )/Gal0x (GK , ∂A~ ), which is a finite group. In such case we have: 0

(a) The invariant field LGalx (GK ,∂A~ ) is the relative algebraic closure K◦ of K in L. (b) K ⊂ K◦ is an algebraic Galois extension of Galois group Gal1x (GK , ∂A~ ). (c) Galy(GK◦ , ∂A~ ) = Gal0x (GK , ∂A~ ). Thus, we can set out: Proposition 4.2.9 K is relatively algebraically closed in L if and only if its Galois group is connected.


117

Galois Correspondence and Group Morphisms Here, we relate the Galois correspondence and the projection of automorphic vector fields through algebraic group morphisms. It is self evident that a ¯ sends an automorphic system A ~ in G with group morphism π : G → G ~ in G ¯ with coefficients in K. coefficients in K to an automorphic system π(A) ~ Furthermore we know that π(A) is an automorphic system in the image of ¯ By restricting our analysis to this image, we π which is a subgroup of G. can assume that π is a surjective morphism. ¯ be a surjective morphism of algebraic groups, Theorem 4.18 Let π : G → G ~ the projected automorphic system π(A). ~ Then: and B ¯ K , ∂ ~ ). (1) y = π(x) is a closed differential point of Diff(G B (2) κ(y) is a strongly normal intermediate extension of K ⊂ κ(y) ⊂ L. ¯ K , ∂ ~ ) = Galx(GK , ∂ ~ )/(ker(π) ∩ Galx(GK , ∂ ~ )). (3) Galy(G A A B (4) Let z be a Kolchin closed point of (Gκ(y) , ∂A~ ) in the fiber of x. Then Galz(Gκ(y) , ∂A~ ) = ker(π) ∩ Galx(GK , ∂A~ ) ¯ K , ∂ ~ ) adherent to y. Then π −1 (x) Proof. (1) Let s be a closed point of Diff(G B is a closed subset of Diff(GK , ∂A~ ) and it contains a closed point z. G(C) acts transitively in the set of closed points, and then there is τ ∈ G(C) such as x = z · τ . Thus, y = s · π(τ ), so that y is closed, s = x, and furthermore ¯ K , ∂ ~ ). π(τ ) ∈ Galy(G B (2) π ♯ : κ(y) → L is a differential K-algebra morphism, and κ(y) is realized as an intermediate extension K ⊂ κ(y) ⊂ L. It is a strongly normal if and only if the subgroup of Gal(L/K) fixing κ(y) is a normal subgroup. We identify Gal(L/K) with Galx(GK , ∂A~ ). Then τ fixes κ(y) if and only if π(τ ) = e. This subgroup fixing κ(y) is ker(π) ∩ Galx(GK , ∂A~ ). By hypothesis, ker(π) is a normal subgroup of G, and then its intersection with Galx(GK , ∂A~ ) is a normal subgroup. Finally, be obtain (3) and (4) by Galois correspondence.

4.2.5

2

Lie Extension Structure on Intermediate Fields

Differential field approach to Lie-Vessiot systems was initiated by K. Nishioka, in terms of the notions of rational dependence on arbitrary constants

118


and Lie extensions (see definitions 3.1.13 and 3.1.14). Here we relate our results with these notions. Theorem 4.19 Assume one of the following: (a) K is algebraically closed. (b) The Galois group of (GK , ∂A~ ) is G. Let y be a particular solution of (MK , ∂X~ ) with coefficients in a differential field extension K ⊂ R. Assume that R is generated by y. Then: (i) K ⊂ R depends rationally on arbitrary constants. (ii) K ⊂ R is a Lie extension. Proof. (i) R is an intermediate extension of the splitting field of the automorphic system which is a strongly normal extension. It is a stronger condition than the one of Definition 3.1.13, thus R depends rationally on arbitrary constants. (ii) If K is algebraically closed, then the result comes directly from Theorem 3.6. For the case (b), some analysis on the infinitesimal structure of R is must be done. If the Galois group is G, then there are not non-trivial differential points in GK , nor in MK . Then R coincides with M(MK ), the field of meromorphic functions in MK . Fundamental vector fields of the action of G on M induce derivations of the corresponding fields of meromorphic functions so that we have a Lie algebra morphism, R(G) → DerK (R),

~ i 7→ X ~ i, A

and the derivation in ∂ in R is seen in M(R) as the Lie-Vessiot system ∂¯ = ∂ +

r X

~ i. fi X

i=1

From that, we have that, ¯ R(G)] ⊂ R(G) ⊗C K, [∂,


119

~ i span the tangent vector space to M , we and because the vector fields X have that the morphism, R(G) ⊗C R → DerK (R) is surjective. According to Definition 3.1.14 we conclude that R is a Lie extension. 2 Remark 4.2.3 In fact, hypothesis (b) is much stronger than we need. But we can not prove a more general result until we have developed Lie-Kolchin reduction (Theorem 5.5). Once we have this technique we can reduce our equation to the Lie algebra of the Galois group. Then we can substitute trivial Galois cohomology of the Galois group for hypothesis (b).

5 Algebraic Reduction and Integration

Here we present the algebraic theory of reduction and integration of algebraic automorphic and Lie-Vessiot systems. Our main tool is an algebraic version of Lie’s reduction method, that we call Lie-Kolchin reduction. Once we have developed this tool we explore different applications.

5.1

Lie-Kolchin Reduction Method

In Chapter 2, when discussing the general topic of analytic Lie-Vessiot systems, we have shown the Lie’s method for reducing an automorphic equation to certain subgroups, once we know certain solution of a Lie-Vessiot associated system. This method is local, because it is assumed that we can choose a suitable curve in the group for the application of the algorithm. A germ of such a curve exists, but it is not true that a suitable global curve exists in the general case. In the algebraic realm we will find obstructions to the applicability of this method, highly related to the structure of principal homogeneous spaces over a non algebraically closed field, and then to Galois cohomology (see [Ko1973], p. 287). We will show that the application of the Lie’s method in the algebraic case leads us directly to Kolchin reduction theorem of a linear differential system to the Lie algebra of its Galois group. Because of this, we decided to use the nomenclature of Lie-Kolchin reduction method.

122

5.1.1

Chapter 5. Algebraic Reduction and Integration

Vanishing of Galois Cohomology

For a formal definition and introduction to Galois cohomology see [Ko1973] and [Se1964]. For an algebraic group G over K the first Galois cohomology set H 1 (G, K) is a pointed set. This set is a group when G is an abelian group. We call trivial cohomology class to the origin of H 1 (G, K). We say that H 1 (G, K) vanish or that it is trivial when it is consists of this trivial cohomology class only. We make a systematical use of the following result which gives us a geometrical interpretation of the Galois cohomology. Thus, we do not need to go deeper in the formal definition. When G denotes an algebraic group over the field of constants C, we write H 1 (G, K) instead of H 1 (GK , K), and the same for any differential extension of K. Theorem 5.1 Let G be an algebraic group over K, and let M be a principal homogeneous G-space. Then M defines a class [M ] ∈ H 1 (G, K). This class classifies M up to K-isomorphism, and reciprocally each element of the Galois cohomology H 1 (G, K) is the class [M ] of a principal homogeneous space. Proof. see [Ko1973] chapter V, section 12, theorem. 10 (p. 281).

2

In particular, the group G is a principal homogeneous G-space. The cohomology class defined by G is trivial. In this way, if H 1 (G, K) is trivial, then all principal homogeneous spaces are isomorphic to G. Note that an algebraic principal homogeneous space over G is isomorphic to G if and only if it has a rational point. In such a case we can state the following. Lemma 5.2 Let G be an algebraic group over K. Then H 1 (G, K) is trivial if and only if any principal homogeneous space over G has a rational point. Proof. If H 1 (G, K) is trivial, then each principal homogeneous space is isomorphic to G, and the identity element e ∈ G is a rational point. Reciprocally, let M be an homogeneous space, and x ∈ M a rational point. Then G ≃ G ×K {x}, and the action of {e} on M gives an isomorphism, G → G ×K {x} → M,

σ 7→ (σ, x) 7→ σ · x.

All principal homogeneous spaces are then isomorphic to G. Thus, to them corresponds the trivial cohomology class. 2

5.1. Lie-Kolchin Reduction Method

123

First Galois cohomology sets of several algebraic groups have been computed by Kolchin, Lang, Serre, and others. In particular we have the following results: (a) H 1 ((K, +), K) is trivial. (b) H 1 ((K∗ , ·), K) is trivial. (c) H 1 (GL(n, K), K) is trivial. (d) H 1 (SL(n, K), K) is trivial. (e) If G is a linear connected solvable group, then H 1 (G, K) is trivial. (f) If K is algebraically closed, H 1 (G, K) is trivial for all algebraic group G. (g) If dim K ≤ 1 (see [Se1964]), then H 1 (G, K) is trivial for any linear algebraic connected group. In particular, extensions of transcendence degree 1 over a characteristic 0 algebraically closed field are fields of dimension less or equal than 1. This happens in the applications when K is the field M(S) of meromorphic functions in a compact Riemann surface S.

5.1.2

Lie-Kolchin Reduction

From now on, let us consider a differential field K of characteristic zero. The field of constant is C, that we assume to be algebraically closed. Let G ~ be an algebraic automorphic vector be an algebraic group over C, and let A field in G with coefficients in K. We also fix a Kolchin closed point x of Diff(GK , ∂A~ ) and denote by L its associated Galois extension. Lemma 5.3 Let G′ ⊂ G be an algebraic subgroup. Assume that a geometric quotient M = G/G′ exist. Then: (a) MK = GK /G′K (b) Let us consider the natural projection morphism πK : GK → MK . For −1 (x) ⊂ GK is an homogeneous space of each rational point x ∈ MK , πK ′ group GK .

124


Proof. (a) C is algebraically closed, and then the geometric quotient is universal; (a) is the fundamental property of geometric universal quotients (see [Sa2001]). (b) The isotropy subgroup Hx of x is certain algebraic subgroup isomorphic and conjugated with G′K . The action of (Hx )K on G preserves −1 (x), the stalk πK −1 −1 (x), (x) → πK ψ : (Hx )K ×K πK the induced morphism −1 −1 −1 (x) (x) ×K πK (x) → πK (ψ × Id) : (Hx )K ×K πK

is the restriction of the isomorphism GK ×K GK → GK ×K GK ,

(τ, σ) 7→ (τ · σ, σ),

and then it is an isomorphism.

2

~ the Lie-Vessiot vector field Let M be an homogenous space over G, and X ~ Let us fix a rational point induced in M by the automorphic vector field A. x0 of M and denote by Hx0 the isotropy subgroup at x0 . Lemma 5.4 Assume that x0 ∈ M is a constant solution of (MK , ∂X~ ). Then: ~ ∈ R(Hx ) ⊗C K. A 0 ~ with coefficients in L such that x0 = τ · x0 . Proof. There is a solution τ of A Therefore τ ∈ (Hx0 )L and its logarithmic derivative is an automorphic vector field in Hx0 , l∂(τ ) ∈ R(Hx0 ) ⊗C L. ~ we obtain A ~ ∈ R(Hx ) ⊗C K. Taking into account that l∂(τ ) = A, 0

2

Theorem 5.5 (Main Result) Let us assume that (MK , ∂X~ ) has a solution x with coefficients in K. If H 1 (Hx0 , K) is trivial, then there exists a gauge ~ to: transformation Lτ of GK that sends the automorphic vector field A ~ = Adjτ (A) ~ + l∂(τ ), B ~ ∈ R(Hx ) ⊗C K an automorphic vector field in Hx . with B 0 0 Proof. Let us consider the canonical isomorphism G/Hx0 → M that sends the class [σ] to σ · x0 . Now, let us consider the base extended morphism, π : GK → MK ,

τ 7→ τ · x0 .


125

We are under the hypothesis of Lemma 5.3 (b). Therefore the stalk π −1 (x) is a principal homogeneous space of group (HK )x which is a subgroup of GK conjugated to (Hx0 )K . Because of the vanishing of the Galois cohomology, there exist a rational point τ1 ∈ π −1 (x), and then τ1 ·x0 = x. Define τ = τ1−1 . Let us consider the gauge transformation, Lτ : (GK , ∂A~ ) → (GK , ∂B~ )

Lτ : (MK , ∂X~ ) → (MK , ∂Y~ ),

~ is the Lie-Vessiot vector field in M induced by B. ~ We have that where Y ~ is an τ · x = x0 is a constant solution of (MK , ∂Y~ ). By Lemma 5.4, B automorphic field in Hx0 . 2 Proposition 5.1.1 Assume that there is a rational point x0 ∈ M such that ~ Galx(GK , ∂A~ ) ⊂ Hx0 , then there exists a rational solution x ∈ M (K) of X. Proof. Let us consider the fundamental solution σ associated to x. We consider it as an L-point of G, σ : Spec(L) → GK . It is determined by the canonical morphism of taking values in σ, σ ♯ : OGK ,x → L = κ(x). Now, let us consider the projection π : G → M , τ 7→ τ · x0 . It induces a morphism π : GK (L) → MK (L). Let us consider x = π(σ). This point x is an L point of M and then it is a morphism x : Spec(L) → MK . Let x ¯ ∈ MK be the image of x; then x is determined by the morphism x♯ defined by the following composition: ♯

OMK ,¯x S π / OGK ,x SSS EE SSSS E SSSS EEσE♯ S SSSSEE x♯ S")

L

We are going to prove that x is a rational point of MK . Let us consider τ ∈ Galx(GK , ∂A~ ). Therefore we have Rτ (x) = x, and the following diagram

126


is commutative: OMK ,¯x UU 00 II UUUU 00 IIII UUUUUUx♯ UUUU II 00 UUUU I$ UUUU 00 */ 00 OGK ,x L ♯ x♯

| σ 00 || | 00 || 00 || 00 (στ )♯ ||| ♯ 00 || Rτ 00 || | }||

L

For each f ∈ OXK ,¯x , we have x♯ (f ) = Rτ♯ (x♯ (f )). This equality holds for all τ ∈ Hx0 . Hence, x♯ (f ) an element of L that is invariant for any differential K-algebra automorphism of L. In virtue of the Galois correspondence the fixed field of L by the action of Galx(GK , ∂A~ ) is K (Theorem 3.12). Thus, x♯ (f ) ∈ K. 2 Theorem 5.6 Let us consider an algebraic subgroup G′ of G verifying: (1) There exist the geometric quotient M = G/G′ , (2) Galx(GK , ∂A~ ) ⊂ G′ , (3) H 1 (H, K) is trivial. Then there exist a gauge isomorphism Lτ of G with coefficients in K reducing ~ to an automorphic system in H, the automorphic system A ~ = Adjτ (A) ~ + l∂(τ ), B belongs to R(G′ ) ⊗C K. Proof. By Proposition 5.1.1 there exists a rational solution of the Lie-Vessiot ~ Theorem 5.5 says that such a reduction exists. system in M associated to A. 2 Denote by Gal0x (GK , ∂A~ ) the connected component of the identity of the Galois group Galx(GK , ∂A~ ). Corollary 5.7 Let K◦ be the relatively algebraic closure of K in L. Assume that the universal geometric quotient G/Gal0x (GK , ∂A~ ) exists, and


127

H 1 (Gal0x (GK , ∂A~ ), K◦ ) is trivial. Then there is a gauge transformation Lτ , τ with coefficients in K◦ such that ~ = Adjτ (A) ~ + l∂(τ ) B belongs to R(Gal0x (GK , ∂A~ )) ⊗C K◦ . Proof. By means of remark (c) (below Proposition 4.2.8), we know that the Galois group of the automorphic system with coefficients in K◦ is precisely Gal0x (GK , ∂A~ ). We apply then Theorem 5.6. 2 Corollary 5.8 If H 1 (Galx(GK , ∂A~ ), K) is trivial then Galx(GK , ∂A~ ) is connected. Proof. If H 1 (Galx(GK , ∂A~ ), K) is trivial, then we can reduce the automorphic system to an automorphic system in R(Galx(GK , ∂A~ )) ⊗C K. Note that Gal0x (GK , ∂A~ ) and Galx(GK , ∂A~ ) have the same Lie algebra. Therefore the Galois group of the reduced equation is contained in Gal0x (GK , ∂A~ ). 2 The following is an extension of the classical result of Kolchin on the reduction a system of linear differential equations to the Lie algebra of its Galois group [Ko1973] Theorem 5.9 (Kolchin) Assume that the universal geometric quotient G/Galx(GK , ∂A~ ) exists. Let us consider the relative algebraic closure K◦ of K in L. There is a gauge transformation Lτ , τ with coefficients in K◦ , such that, ~ = Adjτ (A) ~ + l∂(τ ) B belongs to R(Galx(GK , ∂A~ )) ⊗C K◦ . Proof. Denote by H the Galois group Galx(GK , ∂A~ ). Let us consider M = G/H, and let us denote by x0 ∈ M the origin which is the class of H ~ be the Lie-Vessiot vector field in M associated to A. ~ In in M . Let Y virtue of Proposition 5.1.1, the canonical projection G(L) → M (L) sends the fundamental solution σ to a solution x of (M, ∂Y~ ) with coefficients in K. Let us consider the projection: π : GK → MK . Lemma 5.3 says that the stalk π −1 (x) is a principal homogeneous space modeled over the group HK . Let us denote by P ⊂ GK such homogeneous

128


space. Note that P is {x}, the closure of x in Zariski topology. We have the isomorphism, ψ : P ×K HK → P ×K P,

(τ, g) → (τ, τ g),

Let τ be a closed point of P . Its rational field κ(τ ) is an algebraic extension of K. We have that x = τ · x0 . Thus, we can apply Lie-Kolchin reduction method. Lτ −1 is a gauge transformation with coefficients in κ(τ ): Lτ − 1 : Gκ(τ ) → Gκ(τ ) , ~ to an automorphic vector field B ~ that sends the automorphic vector field A in H with coefficients in κ(τ ). In order to finish the proof we have to see that κ(τ ) is a subfield of the relative algebraic closure K◦ of K in L. It is enough to see that K ⊂ κ(τ ) is an intermediate differential extension of K ⊂ L. Furthermore, if κ(τ ) is an intermediate differential extension then it coincides with K◦ because of the Galois correspondence. Let us consider then the following base extension and natural projection, Pκ(τ ) = P ×K Spec(κ(τ )),

π1 : Pκ(τ ) → P.

The product Pκ(τ ) is a principal homogeneous space modeled over Hκ(τ ) . Moreover, τ induces a rational point of Pκ(τ ) . Hence, the Galois cohomology cohomology class of Pκ(τ ) is trivial, so that it is isomorphic to Hκ(τ ) as homogeneous space. Pκ(τ ) has as many connected components as Hκ(τ ) . We write it as the disjoint union of its connected components. G Pκ(τ ) = Pi . i∈Λ

For each i ∈ Λ, the restriction Pi → P is an isomorphism of K-schemes, and π1 is a trivial covering. But each Pi is a κ(τ )-scheme, and then each component induces in P an structure of κ(τ )-scheme. Hence we have a realization of κ(τ ) as intermediate extension K ⊂ κ(τ ) ⊂ L. Thus, κ(τ ) = K◦ .

2

Remark 5.1.1 Note that in the case of affine groups, due to Chevalley’s theorem (Theorem B.12, appendix), the hypothesis of existence of geometric quotient in Theorem 5.9 is unnecessary.

5.2. Integrability by Quadratures

5.2

129

Integrability by Quadratures

To integrate an automorphic system by quadratures means to write down a fundamental solution by terms of a formula. This formula should involve the solutions of certain simpler equations. We assume that we have a geometrical meccano to express these solutions. We refer to elements of such a meccano as quadratures. Those simpler equations are like the building blocks of our integrability theory. Depending of which simpler equations we consider as integrable we obtain different theories integrability. In theory of Lie-Vessiot systems the elements of our formulas are the exponential maps of Lie groups and indefinite integrals. From a geometric point of view, it is reasonable to consider automorphic systems in abelian groups as integrable. Let us consider an abelian Lie group G. Then, the exponential map, exp : R(G) → G, is a group morphism, and moreover, R(G) is the universal covering of G. An automorphic equation, n

X d log ~i, fi (t)A (x) = dt i=1

~ i ∈ R(G) A

is integrated by the formula, σ(t) = exp

n Z X i=1

t t0

~i fi (ξ)dξ A

!

.

This formula involves the integral of t dependent functions, and the exponential map of the Lie group. Assuming that we are able of realize these operations a reasonable point of view is to consider al automorphic equations in abelian groups integrable. This assumption is done in [Ve1893.a], and followed in [Br1991]. On the other hand, the algebraic case has a new kind of richness. An abelian Lie group splits in direct product of circles an lines, but an abelian algebraic group can carry a higher complexity, for example in the case of abelian varieties. In such case the exponential map is the solution of the Abel-Jacobi inversion problem. In [Ko1953] Kolchin develops a theory of integrability generalizing Liouville integrability, in which just quadratures in one dimensional abelian groups are allowed. It reduces the case to quadratures in the additive group, the multiplicative group and elliptic curves.

130


5.2.1

Quadratures in Abelian Groups

Automorphic Equations in the Additive Group Let us consider an automorphic equation in the additive group C. The additive group is its own Lie algebra, and the logarithmic derivative is the usual derivative. Thus, the automorphic equations are written in the following form: ∂x = a, a ∈ K. (5.1) Definition 5.2.1 An extension of differential fields K ⊂ L is an integral extension if L is K(b), with ∂b ∈ K. We say that b is an integral element over K. It is obvious that the R Galois extension of equation (5.1) is an integral extension of K, with b = a. The additive group (of a field of characteristic zero) has no algebraic subgroups. Therefore, if a is algebraic over K, then a ∈ K. Hence we have two different possibilities for integral extensions: • b ∈ K, Gal(L/K) = {e}, • b 6∈ K, Gal(L/K) = C.

Automorphic Equations in the Multiplicative Group Let us consider now an automorphic equation in the multiplicative group. For the complex numbers C∗ the exponential map is the usual exponential. In the general case of an algebraically closed field of characteristic zero, we can build the exponential map for C ∗ . However, it does not take values in C ∗ but in a bigger group. We avoid such a construction, and then we consider the exponential just as an algebraic symbol. The logarithmic derivative in C ∗ coincides with the classical notion of logarithmic derivative, K∗ → K,

x 7→

∂x . x

The general automorphic equation in the multiplicative group is written as follows: ∂x = a, a ∈ K. (5.2) x


131

Definition 5.2.2 An extension of differential fields K ⊂ L is an exponential extension if L = K(b), with ∂b b ∈ K. We say that b is an exponential element over K. C ∗ has cyclic finite subgroups. Then, we can obtain exponential extensions that are algebraic. There appears the following casuistic: • Gal(L/K) is the multiplicative group C ∗ if b is transcendent over K. • Gal(L/K) is a cyclic group (Zn )∗ if bn ∈ K for certain n. It means that ∂c there is c ∈ K that nc = a. In such case, bn = c. Reciprocally, any algebraic Galois extension of K with a cyclic Galois group is an exponential extension. Here, it is a an essential point that C is algebraically closed.

Automorphic Systems in Abelian Varieties Abelian varieties provide us examples of non linearizable automorphic systems. For the following discussion, let us assume that the constant field of K is the field of complex numbers C. Let G be a complex abelian variety of complex dimension g. Let us consider a basis of holomorphic differentials ω1 , . . . , ωg , and R A1 , . . . , Ag ,B1 , . . . , Bg a basis of the homology of G, we can assume that Ai ωj = δij . Define the Jacobi-Abel map, ∼

g

G− → C /Λ,

p 7→

Z

p

ω1 , . . . e

Z

p

ωg . e

The exponential map is given by the exponential universal covering of the torus and the inversion of the Jacobi-Abel map. Cg F

FF FF FF F" j / Cg /Λ G

exp

A projective immersion of G in P(C, d), for d big enough, is given by terms of theta functions, z 7→ (θ0 (z) : . . . : θd (z)). Hence there are some homogeneous polynomial constrains {P (θ0 , . . . , θd ) = 0}. The quotient θθji defines a meromorphic abelian function in G (see [Mum1970] Chapter 1, Section 3, p.

132


30). Let us consider affine coordinates in G, xi = vector fields of R(Cg ) to G, X ∂ ∂ 7→ Fij (x1 , . . . , xd ) , ∂zi ∂xj

θi θ0 .

We can project the

Fij (x1 , . . . , xd ) =

∂θj ∂zi θ0

j

−

∂θ0 ∂zi θj

θ02

being Fij abelian functions, and then rational functions in the xj . The automorphic system in Cg X i

ai

∂ , ∂zi

ai ∈ K

is seen in A as a non linear system an A, X x˙ j = ai Fij (x1 , . . . , xd ), {P (1, x1 , . . . xd ) = 0}.

(5.3)

i

If b1 , . . . , bd are integral elements over K such that ∂bi = ai , then the solution of the automorphic system (5.3) is: xj =

θj (b) , θ0 (b)

(θ0 (b) : . . . : θd (b)) .

Definition 5.2.3 A strongly normal extension K ⊂ L whose Galois group is an abelian variety is called an abelian extension. For an automorphic system in an abelian variety A we have that the Galois group is an algebraic subgroup of A. Then its identity component is an abelian variety. The Galois extension is then, K ⊂ K◦ ⊂ L, being K◦ ⊂ L an abelian extension.

Example. Let us consider an algebraically completely integrable hamiltonian system in the sense of Adler, Van Moerbecke and Vanhaecke (see [AMV2002]) {H, H2 , . . . , Hn } in C2n . Assume that {Hi (x, y) = hi } are the equations of the affine part of an abelian variety G. The Hamilton equations, x˙ i =

∂H , ∂yi

y˙ i = −

∂H , ∂xi

Hi (x, y) = hi

(5.4)

~ in G with constant coefficients K = C. In are an automorphic system H the generic case, G is a non-resonant torus, and then it is densely filled by


133

a solution curve of the equations (5.4). We conclude that (G, ∂H~ ) has not proper differential points: its differential spectrum consist only of the generic point. In such case, the Galois extension of the system is C ⊂ M(G), the field of meromorphic functions in G.

Automorphic Systems in Elliptic Curves Let us examine the case of an elliptic curve E over C. Assume that E is given as a projective subvariety of P(2, C) in Weierstrass normal form. t0 t22 = 4t31 − g2 t20 t1 − g3 t30 We take affine coordinates x = generated by the vector field, ~v = y

t1 t0

and y =

t2 t0 .

The Lie algebra R(E) is then

∂ ∂ + (12x2 − g2 ) ∂x ∂y

Every automorphic vector field in E with coefficients in K is written in the form a~v with a ∈ K. A solution of the automorphic equation is a point of E with values in the Galois extension L. Such solution have homogeneous coordinates (1 : ξ : η) such that η = a−1 ∂ξ, and ξ is a solution of the single differential equation, (∂ξ)2 = a2 (4ξ 2 − g1 ξ − g2 ).

(5.5)

If we know a particular solution b of (5.5) then we can write down the general solution (1 : ξ : η) of the automorphic equation by means of the addition law in E (see [Ko1953] p. 804 eq. 9), depending of an arbitrary point (1 : x0 : y0 ) ∈ E(C): Sol(5.5) × E(C) → E(L),

(b, (1 : x0 : y0 )) 7→ (1 : ξ : η)

ξ(x0 , y0 ) = −b − x0 − η(x0 , y0 ) = −

1 4

∂b − ay0 a(b − x0 )

∂b + ay0 6 ∂b − ay0 1 + (b + x0 ) − 2a 2 a(b − x0 ) 4

2 ∂b − ay0 a(b − x0 )

(5.6) 3

.

(5.7)

134


Definition 5.2.4 Let K ⊂ L a differential field extension. We say that b ∈ L is a Weierstrassian element if there exist a ∈ K, and g1 , g2 ∈ C, with the polynomial 4x3 − g1 x − g2 having simple roots and such that, (∂b)2 = a2 (4b2 − g1 x − g2 ). The differential extension K ⊂ K(b, ∂b) is called an elliptic extension. The Galois extension of the automorphic equation (5.5) is an elliptic extension of K. It can be transcendent or algebraic. If it is transcendent then its Galois group is the elliptic curve E, if it is algebraic then its Galois group is a finite subgroup of E. Remark 5.2.1 Let us examine the case of complex numbers: assume that the field of constants of K is C. The solution of Weierstrass equation is the elliptic function ℘, and it gives rise to the universal covering of E, π : C → E,

z 7→ (1 : ℘(z) : ℘′ (z)).

The automorphic vector field a~v in E is the projection of the automorphic ∂ in C. The solution of the equation in the additive group is vector field a ∂z R given by an integral element a. Then theRa solution of the projected system R R ′ in E is (1 : ℘( a) : ℘ ( a)). Then b = ℘( a) is the Weierstrass element of the Galois extension. Formulas (5.6) and (5.7) are the addition formulas for the Weierstrass ℘ and ℘′ functions. Example. We obtain the previous situation in the case of one degree of freedom, algebraic complete integrable hamiltonian systems. Let us consider the pendulum equation: y2 x˙ = y − cos(x) = h (5.8) y˙ = sin(x) 2 It is written as a simple ordinary differential equation depending of the energy parameter h, 2 dx = 2h + 2 cos(x), dt by setting z = eix , we obtain the algebraic form of such equation, which is an automorphic equation in an elliptic curve for all values of h except for h = ±1; 2 dz = −z 3 − 2hz 2 − 1. dt


135

The Weierstrass normal form is attained by setting u =

du dt

2

h2 = 4u − u − 3 3

h3 1 + 27 16

−z 4

− 16 h;

.

Hence, the general solution is written in terms of the ℘ functions of invariants 3 2 1 , for h 6= ±1: g2 = h3 and g3 = h27 + 16 2 z(t) = −4℘(t + t0 ) − h ; 3

5.2.2

4h + 3πi x(t) = log −4℘(t − t0 ) − . 6

Liouville and Kolchin Integrability

Definition 5.2.5 Let K ⊂ F a differential field extension. Let us break it up into a tower of differential fields: K = F0 ⊂ F1 ⊂ . . . ⊂ Fd = L. We say that K ⊂ F is: (1) A Liouvillian extension if the differential fields Fi can be chosen in such way that Fi ⊂ Fi+1 is an algebraic, exponential or integral extension. (2) A strict-Liouvillian extension if the differential fields Fi can be chosen in such way that Fi ⊂ Fi+1 is an exponential or integral extension. (3) A Kolchin extension the differential fields Fi can be chosen in such way that Li ⊂ Fi+1 is algebraic, elliptic, exponential or integral extension. Liouvillian and strict-Liouvillian extensions are Picard-Vessiot extensions. An elliptic curve can not be a subquotient of an affine group. Hence, if K ⊂ F is a Kolchin extension and Gal(F/K) is an affine group, then it is a Liouville extension. From this perspective, the following classical result is almost self evident: Theorem 5.10 (Drach-Kolchin) Let K be a field of meromorphic functions of the complex plane C. Assume that the Weierstrass’s ℘ function is not algebraic over K. Then ℘ is not the solution of any linear differential equation with coefficients in K.

136


Proof. Let us assume that this equation exist, and let K ⊂ F na associated its Galois extension. Its Galois group Gal(F/K) is an affine group. We have an intermediate extension: K ⊂ K(℘, ℘′ ) ⊂ F, This intermediate extension K ⊂ K(℘, ℘′ ) is strongly normal and its Galois group is an elliptic curve. Thus, there is a normal subgroup H Gal(F/K) and an exact sequence, 0 → H → Gal(F/K) → E → 0 but the quotient group of an affine group is an affine group, and then E is affine. 2 From the Galois correspondence and some elemental properties of algebraic groups we also have immediately the characterization of Liouvillian and Kolchin extensions in terms of their Galois groups. Proposition 5.2.1 Let K ⊂ L be a strongly normal extension. (1) K ⊂ L is a Kolchin extension if and only if there is a sequence of normal subgroups in Gal(L/K), H0 H1 . . . Hn = Gal(L/K), such that dimC Hi /Hi+1 ≤ 1. (2) K ⊂ L is a strict-Liouville extension if and only if Gal(L/K) is an affine solvable group. (3) K ⊂ L is a Liouvillian extension if and only if the identity component Gal0 (L/K) is a linear solvable group. Proof. For (1) and (3) see [Ko1953]. Let us proof that linear solvable Galois group implies strict Liouville. Let us consider a resolution of the Galois group H0 . . . Hn such that each quotient Hi+1 /Hi is a cyclic group, a multiplicative group or an additive group. This resolution exist by means of Lie-Kolchin theorem. This resolution split the extension K ⊂ L in a tower of differential fields Kn ⊂ Kn−1 ⊂ . . . ⊂K0 .,


137

Each differential extension of the tower is an exponential, integral or algebraic extension with cyclic Galois group. But an algebraic extension with cyclic group is a radical extension. The field C is algebraically closed, hence √ such radical extension is generated by the radical n a of a non-constant element of a, and then it is the Picard-Vessiot extension of the equation, ∂x =

∂a x, na

which is an exponential extension.

2

Integration by Quadratures in Solvable Groups Let us remind that along this chapter we are considering an automorphic ~ with coefficients in K in an algebraic group G defined over C. vector field A We also consider a Kolchin closed differential point x ∈ Diff(GK , ∂A~ ) and the associated Galois extension K ⊂ L. We are going to explain the classical integration by quadratures in terms of Lie-Kolchin reduction method and Galois correspondence. ¯ = G/H. Let us consider a normal subgroup H G, and the quotient group G ¯ K of x. In virtue of Theorem 4.18 we know that, Let y be the projection in G K ⊂ κ(y) ⊂ L, is an intermediate strongly normal extension. Furthermore, the Galois group in y of the automorphic system with coefficients if κ(y) is the intersection of the Galois group Galx(GK , ∂A~ ) with H. Theorem 5.11 Assume that there is a resolution of G, H0 H1 . . . Hn = G, such that dimC Hi /Hi+1 = 1, then K ⊂ L is a Kolchin extension. ¯ i = Hn−i+1 /Hn−i . They are algebraic Proof. Let us consider the quotients G groups of dimension one. Each Gi is isomorphic to one of the following: the additive group, the multiplicative group, or an elliptic curve. Each one corresponds to an integral, exponential, or Weierstrassian quadrature. We prove the theorem by induction in the length of the resolution. Let us consider the projection π : G → G/Hn−1 . Define y = π(x) and let K1 be the relative algebraic closure of κ(x) in L. Then K ⊂ κ(y) is an integral,

138


exponential or elliptic extension and κ(y) ⊂ K1 is an algebraic extension. Hence, K ⊂ K1 is a Kolchin extension. Let z be a closed differential point of (GK1 , ∂A~ ) in the fiber of x. By Theorem 4.18 Galz(GK1 , ∂A~ ) ⊂ Hn−1 , and then by Theorem 5.9 there is a gauge transformation Lτ with coefficients in K1 reducing the automorphic field to an automorphic field in Hn−1 . Any Galois extension associated to this last equation is K1 -isomorphic to L. By the induction hypothesis the extension K1 ⊂ L is a Kolchin extension, hence K ⊂ L is a Kolchin extension. 2 Theorem 5.12 Assume that G is affine and solvable. Then K ⊂ L is a strict-Liouville extension. Proof. The Galois group is a subgroup of G, and then it is a solvable group. The result comes from Proposition 5.2.1 (2) together with Theorem 5.11. 2 Proposition 5.2.2 If there is a connected affine solvable group H ⊂ G such that Galx(GK , ∂A~ ) ⊂ H, then K ⊂ L is a strict-Liouville extension. Proof. H is connected affine solvable an then it has trivial Galois cohomology. We can reduce to the group H by means of theorem 5.6. Hence, we are in the hypothesis of theorem 5.12. 2

5.2.3

Linearization

Reduction by means Chevalley-Barsotti-Sancho Theorem In virtue of Chevalley-Barsotti-Sancho theorem (B.7 in appendix B), there is a unique linear normal connected algebraic group N G such that the quotient G/N and is an abelian variety V . Let us consider the projection ~ be the projected automorphic system π(A) ~ in V , and π : G → V . Let B denote by y the image of x by π. We state the following: Theorem 5.13 Let M be the field of meromorphic functions in VK . Assume that Galy(VK , ∂B~ ) = V , and one of the following hypothesis: (1) H 1 (N, M) is trivial. (2) K is relatively algebraically closed in L.


139

Then, there is a gauge transformation of G with coefficients in M reducing ~ to N . the automorphic system A ~ as an automorphic vector field in G with coefficients Proof. Let us consider A in M. By Galois correspondence we have: Gal(L/M) ≃ Galx(GK , ∂A~ ) ∩ N. If hypothesis (1) holds, then the statement is a particular case of Theorem 5.6. Let us prove the result in the case of hypothesis (2). By Theorem 5.9 there exists a gauge transformations whose coefficients are algebraic over M. By hypothesis Galx(GK , ∂A~ ) is connected. This group Galx(GK , ∂A~ ) realizes itself as a principal bundle over V whose structural group os Gal(L/M). It implies that Gal(L/M) is also connected. So that M is relatively algebraically closed in L. The coefficients of the considered gauge transformation are in M, as we wanted to prove. 2

Linearization by means of Adjoint Representation We consider GL(R(G)) the group of C-linear automorphisms of the Lie algebra R. It is an algebraic group over C. The adjoint representation Adj : G → GL(R(G)) is a morphism of algebraic groups. It gives us a linearization of the equations. Let us consider the center Z(G) and the exact sequence: 0 → Z(G) → G → GL(R(G)) → 0 ~ the projection of the automorphic vector field A ~ by the morDenote by B phism Adj. It is a linear system and then its Galois extension K ⊂ P is a Picard-Vessiot intermediate extension of K ⊂ L. Proposition 5.2.3 P ⊂ L is a strongly normal extension and Gal(L/P) is an abelian group. ~ with coefficients in Proof. The extension P ⊂ L is a Galois extension of A P, so that it is strongly normal. Its Galois group is, by the Galois correspondence, the intersection of the Galois group of Galx(GK , ∂A~ ) with the center Z(G); it is an abelian group. 2

140


Linearization by means of Global Regular Functions The ring of global regular functions Γ(OG , G) is a Hopf algebra, and then it spectrum is a linear algebraic group L = Spec(Γ(OG , G)). The kernel C of the canonical morphism π : G → L is, by definition a quasi-abelian variety (see [Sa2001]). Let us consider the exact sequence: 0 → C → G → L → 0. We proceed as we did in Proposition 5.2.3, and then we obtain the following result. Proposition 5.2.4 Let K ⊂ P be the Picard-Vessiot extension of the auto~ in L. Then P ⊂ L is a strongly normal extension, and morphic system π(A) the connected component of the identity of its Galois group is a quasi-abelian variety.

5.3

Integrability of Linear Equations

This section is devoted to the Liouville integrability of linear differential equations. Since the development of Picard-Vessiot system it is a rich field of research, let us cite some important specialized literature [Kov1983], [Si-Ul1993.a], [Si-Ul1993.b], [Ul-We1994], [Ho-We1997], [HRUW1999]. Here, we adopt a slightly different point of view on linear differential equations. We see them as automorphic systems. It gives us some insight into the geometric mechanisms that allows quadratures. In this way we are able to measure the solvability of the Galois groups, in terms of equations in flag varieties and grassmanians (Theorem 5.14). They are the natural geometrical generalization of Riccati equations. From now on let G be a linear connected algebraic group over C. We consider ~ an automorphic vector field in G with coefficients in K. A

5.3.1

Flag Variety

We call Borel subgroup of G to any maximal connected solvable group of G. Borel subgroups are all conjugated and isomorphic subgroups. The quotient space G/B is a complete variety (see [Sa2001] p. 163, th. 10.2).

5.3. Integrability of Linear Equations

141

Definition 5.3.1 We call flag variety of G to the homogeneous space quotient G/B, being B a Borel subgroup of G. The flag variety of G is defined up to isomorphism of G-homogeneous spaces. Let us consider F lag(G) a flag variety of G, and let (F lag(G), ∂F~ ) be the induced Lie-Vessiot system. Let us see a natural generalization of the well-known theorem of J. Liouville that relates the integrability by Liouvillian functions of the second order linear homogeneous differential equation with the existence of an algebraic solution of an associated Riccati equation. This classical result is the particular case of GL(2, C) in the following general Liouville’s theorem. Theorem 5.14 The Galois extension K ⊂ L is Liouvillian if and only if the flag Lie-Vessiot system (F lag(G), ∂F~ ) has an algebraic solution with coefficients in K◦ , the algebraic relative closure of K in L. Proof. By the Galois correspondence we have that the Galois group of (GK◦ , ∂A~ ) is the connected identity component of the Galois group of (GK , ∂A~ ). Assume that (F lag(G), ∂F~ ) has an algebraic solution x ∈ F lag(G)(K◦ ). We are under the hypothesis of Theorem 5.9. There is a gauge transformation of ~ to an automorphic vector field B ~ in the Borel subgroup B. GK0 that send A ~ with coefficients in K0 is contained in a Borel Then the Galois group of B subgroup. Then the connected component of Galx(GK , ∂A~ ) is solvable. Reciprocally, let us assume that K ⊂ L is a Liouvillian extension. In such case the identity connected component of the Galois group is contained in a Borel subgroup B. By Proposition 5.1.1 there is a solution with coefficients in K◦ of F~ . 2

5.3.2

Automorphic Equations in the General Linear Group

Grassmanians Let us consider E as n-dimensional vector space. Along this text m-plane will mean m-dimensional linear subspace. For all m ≤ n the linear group GL(E) acts transitively in the set of m-planes. For an m-plane Em , the stabilizer subgroup is an algebraic group, and then the set of m-planes define an algebraic homogeneous space.

142


Definition 5.3.2 We call grassmanian of m-planes of E, Gr(E, m), to the homogeneous space whose closed points are the m-planes of E. Denote Gr(C, n, m) the grassmanian of m-planes of C n . Example. Gr(C, n, 1) is the space of lines in C n , and then if its the projective space of dimension n − 1, P(n − 1, C). The Gr(C, n, n − 1) is the space os hyperplanes and then it is the dual projective space P(n − 1, C)∗ . In general, m-planes of E are in one-to-one correspondence with (n − m)planes of the dual space E ∗ , and then we have the projective duality Gr(E, m) ≃ Gr(E ∗ , n − m). The action of GL(E) on Gr(E, m) is not faithful. Each scalar matrix of the center of GL(C, n) fix all m-planes. Thus, the non faithful action of GL(E) is reduced to a faithful action of the projective group P GL(E). All grassmanian are projective varieties. There is a canonical embedding of n Gr(E, m) into the projective space of dimension (m ) − 1, called the pl¨ ucker embedding: Gr(E, m) → P(E ∧n ),

he1 , . . . , em i 7→ he1 ∧ e1 ∧ . . . ∧ em i.

For computation in the grassmanian spaces we will use pl¨ uckerian coordinates. This system of coordinates is subordinated to a basis in E. Thus, let us consider a basis {e1 , . . . , en }. Let E1 = he1 , . . . , em i be the m plane spanned by the first m elements of the basis, and define E2 = hem+1 , . . . en i its complementary. Let us consider the projection π : E → E2 of kernel E1 . We define the open subset U ⊂ Gr(E, m), U = {F : F ⊕ E2 = E}. For F ∈ U the splitting of the space induces an isomorphism iF : E1 → F . We have an isomorphism ∼

U− → HomC (E1 , E2 ),

F 7→ π ◦ iF .

We define the pl¨ ukerian coordinates of F as the matrix elements of π ◦ iF in the above mentioned basis. By permuting the elements of the basis we n construct a covering of Gr(E, m) by (m ) affine open subsets isomorphic to C n(n−m) .


143

Let us compute pl¨ uckerian coordinates in Gr(C, m, n) related to the canonical basis. Let us consider F ∈ Gr(C, m, n), and a basis of F , {~x1 , . . . , ~xm }, ~xi = (x1i , . . . , xni ). The matrix,   x11 . . . x1m  x21 . . . x2m     .. ..  . .  . . .  xn1 . . . xnm

is of maximal rank. Thus, there is a non vanishing minor of rank m. In particular, F is in the open subset U if and olny if the minor corresponding (m) to the first m rows does not vanish. In such case we define the numbers λij 

x11  x21   ..  .

xn1



1 .. .

... .. . ... ... .. .

0 .. .



    −1  . . . x1m    x11 . . . x1m    . . . x2m   . 0 1  ..  =  .. .    (m) (m)   .. . .. . . λ λ    . . 11 1m   . xm1 . . . xmm .   .. . . . . xnm   . (m) (m) λn−m,1 . . . λn−m,m

that are the pl¨ uckerian coordinates of Em ∈ Gr(C, m, n) in the open affine subset U related to the split of C n as E1 ⊗ E2 .

Flag Variety of the General Linear Group A flag of subspaces of C n , is a sequence, E1 ⊂ E2 ⊂ . . . ⊂ En−1 ,

dimC Ei = i

of linear subspaces of C n . The space F lag(C, n) of flags of C n is an homogeneous space of GL(C, n), and it is faithful for the action of P GL(C, n). There is a canonical morphism, F lag(C, n) →

n−1 Y

m=1

Gr(C, n, m),

E1 ⊂ E2 ⊂ En−1 7→ (E1 , . . . , En−1 ).

By Lie-Kolchin theorem the isotropy subgroup of a flag is also a Borel subgroup. Then, we can state F lag(C, n) is the flag variety of the general linear

144


group. Let us introduce a system of coordinates in F lag(C, n). Let us consider {e1 , . . . , en } the canonical basis of C n . Each σ ∈ GL(C, n) defines a flag F (σ) as follows: hσ(e1 )i ⊂ hσ(e1 ), σ(e2 )i ⊂ . . . ⊂ hσ(e1 ), . . . , σ(en−1 )i. There is a canonical flag corresponding to the identity element. Its isotropy group is precisely T (C, n) the group of upper triangular matrices. Then two matrices A, B ∈ GL(C, n) define the same flag if and only if A = BU for certain U ∈ T (C, n). Then let us consider the affine subset of GL(C, n) of matrices with non vanishing principal minors. For such a matrix there exist a unique LU decomposition such that U ∈ T (C, n) and is a lower triangular matrix as follows,   1 0 ... 0  λ21 1 . . . 0   A= . ..  U .. . . .  . . . . λn1 λn2 . . . 1

Hence the matrix elements λi define a system of affine coordinates in F lag(C, n), in certain affine open subset. We construct an open covering of the flag space by permutating the vectors of the canonical base. The canonical morphism Y F lag(C, n) → Gr(C, m, n) m

is easily written in pl¨ uckerian coordinates: (m)

λij

= λi+m,j −

m X

λi+m,k λkj .

k=1

Matrix Riccati Equations Let us consider an homogeneous linear differential equation x˙ = Ax,

A ∈ gl(K, n).

It is seen as an automorphic system that induces Lie-Vessiot systems in each homogeneous space. Let us compute the induced Lie-Vessiot systems in the grassmanian spaces. First, the linear system induces a linear system in (C n )m . X˙ = AX, (5.9)


145

where X is a n × m matrix. We write X = YU , being U a m × m matrix and Y a (n − m) × m matrix. Λm = Y U −1 is the matrix of pl¨ uckerian coordinates of the space generated by the m column vectors of the matrix X. Then, Λ˙ m = Y˙ U −1 − Λm U˙ U −1 . If we decompose the matrix A in four submatrices A11 A12 A= A21 A22 being A11 of type m × m, A12 of type m × (n − m), A21 of type (n − m) × m, and A22 if type m × m. Them the matrix linear equation (5.9) splits as a system of matrix linear differential equations, U˙ = A11 + A12 Y,

Y˙ = A21 U + A22 Y,

from which we obtain the differential equation for affine coordinates in the grassmanian, Λ˙ m = A21 + A22 Λm − Λm A11 − Λm A12 Λm

(5.10)

which is a quadratic system. We call such a system a matrix Riccati equation associated to the linear system.   (m) (m) λ11 ... λ1,m  .  .. ..  . Λm =  . . .   (m) (m) λn−m,1 . . . λn−m,m (m) λ˙ ij = am+i,j +

n−m X k=1

(m)

am+i,m+k λkj −

m X k=1

(m)

λik akj −

X

(m)

(m)

λik ak,r+mλrj

k=1...m r=1...n−m

Example. Let us compute the matrix Riccati equations associated to the general linear system of rank 2 and 3. First, let us consider a general linear system of rank 2, x˙ 1 = a11 x1 + a12 x2 ,

x˙ 2 = a21 x1 + a22 x2 .

There is one only grassmanian Gr(C, 1, 2), which is precisely the projective line. The associated matrix Riccati equation is an ordinary Riccati equation x˙ = a21 + (a22 − a11 )x − a12 x2 .

146


In the case of a general system of rank 3,      x˙ 1 a11 a12 a13 x1 x˙ 2  = a21 a22 a23  x2  x˙ 3 a31 a32 a33 x3 there are two grassmanian spaces, Gr(C, 1, 3) and Gr(C, 2, 3), being the projective plane P2 (C) and the projective dual plane P2 (C)∗ respectively. Then we obtain two quadratic systems, x˙ = a21 + (a22 − a11 )x + a23 y − a12 x2 − a13 xy P(2, C) y˙ = a31 + (a33 − a11 ) + a32 x − a13 y 2 − a12 xy ∗

P(2, C)

ξ˙ = a31 + (a33 − a11 )ξ + a21 η − a23 ξη − a13 ξ 2 η˙ = a32 + (a33 − a22 )η + a12 ξ − a13 ξη − a23 η 2

called the associated projective Riccati equations.

Flag Equation From the relation between pl¨ uckerian coordinates and affine coordinates in the flag variety we can deduce the equations of the induced Lie-Vessiot system in F lag(C, n), from the matrix Riccati equations. We will obtain a Riccati quadratic equation for n = 2, and a cubic system for n ≥ 3. λ˙ ij = aij +

n X

k=j+1

−

j n X X

aik λkj −

λik akr λrj +

k=1 r=j+1

j X

λik akj +

j j X X

λir λrk akj

k=1 r=k+1

k=1

j n X X

j X

λis λsk akr λrj ,

k=1 r=j+1 s=k+1

Setting λii = 1 for all i, we can simplify these equations. λ˙ij =

n X k=j

aik λkj −

j X n X k=1 r=j

λik akr λrj +

j j n X X X

λir λrk aks λsj

(5.11)

k=1 r=k+1 s=j

Such as cubic system can be seen as a hierarchy of projective Riccati equations. The equation corresponding to the first column λi1 , i = 2 . . . , n is a projective Riccati equation in P(n − 1, C). The equation corresponding to


147

the second column is a projective Riccati equation in P(n − 2, C(λi1 )), and so on. Example. Let us compute the flag equation for the general differential linear system of rank 3. Denote x = λ21 , y = λ31 , z = λ32 . x˙ = a21 + (a22 − a11 )x + a23 y − a12 x2 − a13 xy y˙ = a31 + a32 x + (a33 − a11 )y − a12 xy − a13 y 2 z˙ = a32 − a12 y + (a33 − a22 + a12 y − a13 y)z + (a13 y − a23 )z 2

5.3.3

(5.12)

Equations in the Special Orthogonal Group

Automorphic equations in special orthogonal group have been deeply studied since 19th century [Ve1893.b], [Da1894]. In particular Darboux related these equation with Riccati equation. He stated that the integration of (5.13) is reduced to the integration of (5.19). Here we show that the Flag equation of an automorphic equation in SO(C, 3) is precisely the Riccati equation, and then the solutions of (5.13) are Liouvillian if and only if there are algebraic solutions for (5.19) The Lie algebra so(3, C) is the algebra of skew-symmetric matrices of gl(C, 3). Then an automorphic system in SO(3, C) is written in the following form.      x˙ 0 a b x0 x˙ 1  = −a c x1  a, b, c ∈ K, (5.13) x˙ 2 −b −c x2

where the void spaces represent the vanishing elements in the matrix.

On the Structure of the Special Orthogonal Group The special orthogonal group is the group of linear transformations preserving the quadratic form x20 + x21 + x22 . Let us consider the non degenerated quadric in the projective space S2 ⊂ P(3, C), defined by homogeneous equation {t20 + t21 + t22 − t23 = 0}. In affine coordinates xi = tt3i , its affine part is a sphere of radius 1. Thus SO(3, C) is a subgroup of algebraic automorphisms of the quadric; SO(3) ⊂ Aut(S2 ). Each non degenerate quadric in the projective space over an algebraically closed field is a hyperbolic ruled surface. It has two systems of generatrices,

148


being each system parameterized by a projective line. Denote P1 , P2 these projective lines. p ∈ P1 , and q ∈ P2 are lines S2 , and they intersect in a unique point s(p, q) ∈ p ∩ q. We have a decomposition of S2 which is a particular case of Segre isomorphism, ∼

P1 ×C P2 − → S2 ⊂ P(3, C)  t0 = u0 v1 + u1 v0    t1 = u1 v1 − u0 v0 ((u0 : u1 ), (v0 : v1 )) 7→ (t0 : t1 : t2 : t3 ) t = i(u1 v1 + u0 v0 )   2 t3 = u0 v1 − u1 v0

Let us consider any algebraic automorphism of S2 . τ : S2 → S2 . In particular, it must carry a system of generatrices to a system of generatrices. Let us denote P1 , P2 to the two system of generatrices of S2 . Hence, τ is induces by a pair of projective transformations (τ1 , τ2 ), where τ1 : P1 → P1 ,

τ2 : P2 → P2

τ1 : P1 → P2 ,

τ2 : P2 → P1 .

or We conclude that the group of automorphism of S2 is isomorphic to the following algebraic group, Aut(S2 ) = P GL(1, C) ×C P GL(1, C) ×C Z/2Z. Let us compute the image of the canonical monomorphism SO(3, C) ⊂ Aut(S2 ). We take affine coordinates in the pair of projective lines, x = uu01 , y = vv10 . This is the system of symmetric coordinates of the sphere introduced by Darboux [Da1894]. x0 =

1 − xy x−y x=

x1 = i

x0 + ix1 1 − x2

1 + xy x−y y=

x2 =

x2 − 1 . x1 − ix2

x+y x−y

Let us write a general element of SO(3, C) in affine coordinates,    µ+µ−1 µ−1 −µ 1 1 2 2i   −1 λ+λ−1 λ−1 −λ   µ−µ−1 ν+ν −1 µ+µ Rλ,µ,ν =   2 2i   2i 2 2 −1 −1 λ−λ ν−ν −1 λ+λ 1 2i 2 2i

(5.14) (5.15)



ν −1 −ν  2i  ν+ν −1 2


149

where, in the complex case λ = eiα , µ = eiβ , ν = eiγ are the exponentials of the Euler angles. Direct computation gives us, ( (λµν+λν+µν−ν+λµ−λ+µ+1)x+λµν+λν+µν−ν−λµ+λ−µ−1 x 7→ (λµν+λν−µν+ν+λµ−λ−µ−1)x+λµν+λν−µν+ν−λµ+λ+µ+1 = rλ,µ,ν (x) Rλ,µ,ν (λµν+λν+µν−ν+λµ−λ+µ+1)y+λµν+λν+µν−ν−λµ+λ−µ−1 y 7→ (λµν+λν−µν+ν+λµ−λ−µ−1)y+λµν+λν−µν+ν−λµ+λ+µ+1 = rλ,µ,ν (y) and then Rλ,µ,ν induces the same projective transformation rλ,µ,ν for x and y. Hence, SO(3) ⊆ P GL(1, C) ⊂ Aut(S2 ). In particular, we have the following formulae for rotations around euclidean axis:   1 (λ + 1)x + (λ − 1)  λ+λ−1 λ−1 −λ  (5.16)  2 2i  : x 7→ (λ − 1)x + (λ + 1) λ−λ−1 λ+λ−1 

2i

2

λ+λ−1

λ−1 −λ

 λ−λ2 −1  2i

2i λ+λ−1 2



1



λ+λ−1  2

λ−λ−1 2i

1



  : x 7→ λx

(5.17)

 : x 7→

(5.18)



λ−1 −λ 2i  λ+λ−1 2

(λ + λ−1 + 1/2)x − i(λ − λ−1 ) i(λ−1 − λ)x − (λ + λ−1 + 1/2)

An the following formulae for the induced Lie algebra morphism – the are computed by derivation of previous formulae with λ = 1 + iε –. Here the Lie algebra pgl(1, C) is identified with sl(2, C):   i 1 −1  7→ 2 −i 2 0   1 1 2   0 7→ − 12 −1   0 − 2i   1 7→ − 2i −1 Reciprocally, a projective transformation x 7→

u11 x + u12 ; u21 x + u22

y 7→

u11 y + u12 , u21 y + u22

150


induces a linear transformation in the affine coordinates x0 , x1 , x2 (see [Da1894] p. 34). SO(C, 3) is precisely the group of automorphisms of S2 that are linear in those coordinates. We have proven the following proposition which is due to Darboux. Proposition 5.3.1 The special orthogonal group SO(3, C) over an algebraically closed field is isomorphic to the projective general group P GL(1, C). The isomorphism is given by formulae (5.16), (5.17), (5.18).

Flag Equation The flag variety of SO(3, C) is a projective line. Any of the Darboux symmetric coordinates, x : S2 → P1 gives us a realization of the action of SO(3) on P1 . By substituting the equation (5.13) in the identities (5.14), (5.15) we deduce the Riccati differential equation satisfied by this symmetric coordinate, which is the flag equation of equation (5.13): x˙ =

−b + ic 2 −b − ic − iax + x . 2 2

(5.19)

In [Da1894], Darboux reduces the integration of the equation (5.13) to finding two different particular solutions of the Riccati equation (5.19). By application of our generalization of Liouville’s theorem we obtain an stronger result. Theorem 5.15 (Darboux) The Galois extension of the equation (5.13) is a Liouvillian extension of K if and only if the Riccati equation (5.19) has an algebraic solution. Proof. It is a particular case of Theorem 5.14.

2

6 Conclusions and Work in Progress

Conclusions Lie-Vessiot equations appear frequently both in the theoretical development and in applied problems. However, this thesis is surely the largest text devoted to this subject, together with [CGM2000]. We guess that it is due to two different facts. First, the Lie’s theorem on superposition has not been well understood until now. Thus, this important result was doomed to the limbus. Second, most Lie-Vessiot equations appearing in applications are linear. Notwithstanding, the results that arise when we adopt the general point of view are interesting, even for the linear case. By rescuing this Lie’s result to the modern language of global geometry, we can follow the steps of our predecessors. Differential equations admitting superposition laws are related to pretransitive actions of Lie groups, as has been defined in chapter 2. Hence, they are equivalent to automorphic systems. This point definitely justifies the connection between superposition laws and automorphic systems. Another interesting point is the advance in the group theoretic foundations of differential Galois theory. In Picard-Vessiot’s original presentation this group theoretical essence is hidden behind the linearity of the differential equations. In Kolchin’s theory of strongly normal extension, the group theoretic point of view emerges again, but the differential equations are hidden. Here, we put together differential equations with a group theoretic approach to differential Galois theory. What we develop is Vessiot’s theory of automorphic equations. Conceptually we do nothing new with respect to Vessiot. However, we have

152

Chapter 6. Conclusions and Work in Progress

access to the modern language of analytic and algebraic geometry. Then we are able to overcome the difficulties that stopped Vessiot. He anticipates our work in [Ve1893.b] and [Ve1894] and finally he performed in [Ve1940] the part that is feasible in the classical setting. We think that the most remarkable fact that we pointed out during our research is the intimate connection between Lie-Vessiot equations and strongly normal extensions of differential algebra. We guess that this connection was implicitly known by Kolchin and other differential algebraists. But it has never became explicit until now. Another important conclusion is the relation between integration by quadratures in Picard-Vessiot theory and Lie’s reduction method. It appears that all known mechanisms for reduction of differential equations come from an application of a Lie’s reduction in somewhere. It is an encouraging fact that we can see how classical results of differential algebra from Liouville, Darboux and Kolchin arise easily when we apply Lie’s reduction in a suitable frame. A group theoretic presentation of differential Galois theory is a suitable frame for studying the relation between the integrability of differential equations and the structure of their groups of symmetries. We have obtained some partial results in this direction. It is expected that, in future, we can unify Galois theories with Lie’s point of view on differential equations. To resume, the conclusion of our research is that the theory of Lie-Vessiot and automorphic systems gives us a more geometrical approach to differential Galois theory. In this context we are able to relate differential Galois theory to others approaches to the integrability of differential equations.

Work in Progress Infinite Dimensional Galois Theories Our differential Galois theory is non-linear but finite dimensional. Vessiot went from finite dimensional theory [Ve1893.a] to infinite dimensional [Ve1904]. Now, there is an growing interest in non-linear differential Galois theories (see [Um1996], [Mlg2001] and [Mlg2002]). We think that the key-point is not the non-linearity but the infinite dimension. Thus, a general non-linear differential equation should be translated to an automorphic

153

system in the infinite dimensional group of automorphisms of the phase space. Then, we should be able to apply the theory of automorphic systems. This is the path followed by Vessiot and recognized by Cartan [Ca1947] and Umemura [Um1997]. Our first aim is to find the equivalent to Kolchin’s theorem on reduction in the infinite dimensional frame.

Symmetries and Galois Theory Athorne studied the differential algebraic structure of the Lie algebra of symmetries of linear differential equations (see [Ath1997]). Throughout this thesis we have generalized some of his results from the linear case to the Lie-Vessiot frame (Theorem 2.21). Some of the work that connects Galois theory with Lie symmetries has already been done. We know that time dependent right invariant transversal symmetries are solutions of an adjoint Lie-Vessiot equation. However, we still miss the nexus between the Galois group and the general Lie algebra of symmetries.

Criteria of Integrability for Non-linear Systems The study of the Lie algebra of symmetries in the general non-linear (infinite dimensional) case seems to be fruitful. Athorne [Ath1998] gave a differential algebraic description of this Lie algebra of symmetries. In the non-linear case, the wider notion of integrability –known by the author– is the concept of solvable structure (see [A-H1994]). It is the natural refinement of Lie’s concept of solvable algebra of symmetries. We think that this concept is highly related to reduction in terms of Galois theory. We expect to find a general theorem in the Morales-Ramis frame (see [Mo-Ra2001.a], [Mo-Ra2001.b]). We conjecture the following. If a non-linear system admits a solvable structure then the variational equations along solutions have virtually solvable Galois groups.

Future perspectives Algebraically Completely Integrable Hamiltonian Systems One important and beautiful link between algebraic geometry and differential equation is the theory of algebraically completely integrable Hamiltonian systems (see [Va1996] and [AMV2002]). In fact we know that al-

154

Chapter 6. Conclusions and Work in Progress

gebraically completely integrable Hamiltonian systems give us examples of non-linearizable differential equations related to strongly normal extensions. In [Ca1947], Cartan said that mechanical systems are interesting examples of infinite dimensional automorphic systems.

Log-Riemann Surfaces and Galois Theory The theory of log-Riemann surfaces recently developed by Perez-Marco and Biswas [PB2004] seems to lead us to amazing results in a crossroad between complex analysis, differential algebra, Galois theories and analytic theory of numbers. We have developed an analytic Galois theory. In the meromorphic case, the solutions of automorphic systems have coefficients that are functions in log-Riemann surfaces. There is a monodromy representation of the group of automorphisms of the log-Riemann surfaces into the analytic Galois group. Under certain conditions we expect this monodromy to be dense in the Galois group. To find those conditions is to extend Schlesinger theorem to the analytic frame. Moreover, it is interesting to relate problems of convergence of log-Riemann surfaces with problems of convergence or bifurcation of differential equations. For example, differential equations: y˙ =

1 1 y+ nx x

(En )

are Lie-Vessiot equations related to the action of the multiplicative group C∗ , λ · y = λy − (λ − 1)n, and its analytic differential Galois group are cyclic groups Z/nZ. They are √ satisfied by n n x − n, which is a function in the n-cyclotomic Log-Riemann surface. When n tends to infinity, equation (En ) tends to the equation of the logarithm, 1 (E∞ ) y˙ = , x which is an automorphic equation in the additive group C, whose analytic Galois group is 2πiZ which is the group of automorphisms of the log-Riemann surface of the logarithm. In these case we have that the n-cyclotomic logRiemann surface tends, in Caratheodory topology, to the log-Riemann surface of the logarithm. Can we relate these facts with differential Galois theory? What kind of topology of differential equations is related to the convergence of Galois groups?

155

Algorithmic Aspects One of the stronger points of Galois theories is that they give us algorithms for solving equations. There is some specialized research in this field; let us mention [Kov1983] [Ho-We1997], [Si-Ul1993.a], [Si-Ul1993.b] and [Ul-We1994], [HRUW1999]. From the point of view of automorphic systems, all information of the Galois group of a linear differential equation is contained in the algebraic solutions of the Flag equation – that includes matrix Riccati equations –. In particular, higher order Riccati equations used for reduction, seems to be a particular case of the projective Riccati equation. Does our framework lead us to new algorithms for reduction of differential equations?

Differential Algebraic Groups and Algebraic D-groups Automorphic systems seems to be a particular case of Pillay’s algebraic D-groups [Pi2004]. Under suitable hypothesis for K: Are all algebraic Dgroups automorphic systems? In such case, Pillay’s differential Galois theory is reduced to Lie-Vessiot theory. In the general case: What is the richness of algebraic D-groups beyond automorphic systems?

A Algebraic Geometry

Along this section we make a review of some fundamental facts in algebraic geometry. Our objective is to set our notation and conventions about scheme theory, and to point out some relevant results and tools of algebraic geometry and commutative algebra. The main reference of this section is [Ha1977], and then indirectly also [EGA].

A.1

Sheaves and Presheaves

Let X be a topological space. Definition A.1.1 A presheaf P of abelian groups on X consists on following the data: (1) for every open subset U ⊆ X, an abelian group P(U ), (2) for every inclusion V ⊆ U of open subsets of X, a morphism of abelian groups ρU V : P(U ) → P(V ), subject to the conditions, (S1) P(∅) = 0, (S2) ρU U is the identity map, (S3) for W ⊆ V ⊂ U , three open subsets, ρU W = ρV W ◦ ρU V . The elements s ∈ P(U ) are called sections of P in U . The morphism ρU V is called restriction morphism, we will write s|V for the restriction ρU V (s).

158

Appendix A. Algebraic Geometry

Definition A.1.2 If P is a presheaf on X, and x ∈ X, we define the stalk of Px to be the direct limit of the groups P(U ) for all open sets U containing x, via the restriction maps, Px = lim P(U ). → x∈U

The elements s ∈ Px are called germs of sections of P around x. For a section s ∈ P(U ), with x ∈ U , we write sx for the germ in x defined by the section s. Definition A.1.3 If P, Q are presheaves on X, a morphism ϕ : P → S consist of a morphism of abelian groups ϕ(U ) for each open set U ⊆ X, such that for all open subset V ⊆ U , the following diagram is commutative: P(U )

ϕ(U )

/ Q(U )

ρU V

ρU V

P(V )

ϕ(V )

/ Q(V )

Note that a morphism ϕ : P → Q induces morphisms between the stalks ϕx : Px → Qx . Definition A.1.4 Let ϕ : P → Q be a morphism of presheaves in X. Then, the following formulae 1. U ; ker(ϕ(U )) 2. U ; coker(ϕ(U )) 3. U ; im(ϕ(U )) define presheaf kernel, presheaf cokernel, and presheaf image of ϕ respectively. Definition A.1.5 A sequence of presheaves morphisms, ϕi−1

ϕi

. . . → Pi−1 −−−→ Pi −→ Pi+1 → . . . is called an exact sequence, if for each open subset U ⊂ X, . . . → Pi−1 (U ) → Pi (U ) → Pi+1 (U ) → . . . , is an exact sequence of abelian groups, or equivalently the kernel presheaf of ϕi coincides with the image presheaf of ϕi−1 .

A.1. Sheaves and Presheaves

159

Definition A.1.6 A presheaf S on X is a sheaf if it satisfies de following additional conditions, (S4) if U is an open set, and {Vi }i∈Λ is an open covering of U , and if s ∈ S(U ) is an element such that s|Vi = 0 for all i ∈ Λ, then s = 0. (S5) if U is an open set, and {Vi }i∈Λ is an open covering of U , and if we have elements si ∈ S(Vi ) for each i ∈ Λ, with the property that for each i,j, si |Vi ∩Vj = sj |Vi ∩Vj , then there is an (unique) element s ∈ S(U ) such that s|Vi = si for all i ∈ Λ. Proposition A.1.1 Given a presheaf P, there is a sheaf P + and a morphism θ : P → P + , with the property that for any sheaf S and any morphism ϕ : P → S there is a unique morphism ψ : P + → S such that ϕ = ψ ◦ θ. Furthermore the pair (P + , θ) is unique up to unique isomorphism. P + is called the sheaf associated to de presheaf P. Proof. We just show the construction of the S associated sheaf. For a complete ˜ proof, see [Ha1977]. Let us consider P = x Px the union of stalks. There is a canonical projection π : P˜ : X. We endow P˜ with the finer topology what does all the functions, s : U → P˜ ,

s : x 7→ sx ,

with s ∈ P(U ), together with the projection π continuous. π : P˜ → X is ˜ the space of called the étale space associated to P. Then P + (U ) = Γ(U, P), continuous mappings s : U → P˜ such that π ◦ s = IdU . 2 Let us consider a morphism of sheaves on X, ϕ : S → T . Then the kernel presheaf of ϕ is a sheaf on X. In general the cokernel and image presheaves are not sheaves on X. Then, Definition A.1.7 We call ker(ϕ) to the kernel sheaf of ϕ, im(ϕ) to the sheaf associated to the image pre-sheaf of ϕ, and coker(ϕ) to the sheaf associated to the cokernel pre-sheaf of ϕ. Definition A.1.8 A sequence of sheaves morphisms, ϕi−1

ϕi

. . . → Si−1 −−−→ Si −→ Si+1 → . . . is called an exact sequence if ker(ϕi ) = im(ϕi−1 ).

160


Remark A.1.1 Note that the condition of being an exact sequence of sheaves is a weaker condition of being an exact sequence of presheaves. It is an important point to notice that an injective morphism of sheaves, 0 → S′ → S induces injective morphisms on sections, 0 → S ′ (U ) → S, but a surjective morphism

S → S¯ → 0

does not in general induce surjective morphisms in sections. Theorem A.1 A sequence of sheaves morphisms ϕi−1

ϕi

. . . → Si−1 −−−→ Si −→ Si+1 → . . . is exact, if and only if the sequence of abelian groups induced in the stalk, ϕi−1

ϕi

. . . → Si−1,x −−−→ Si,x −→ Si+1,x → . . . is exact for all x ∈ X.

Continuous Maps Let f : X → Y be a continuous map. Definition A.1.9 Let S be a sheaf in X. We define the direct image f∗ S as a sheaf in Y by setting, f∗ S(V ) = S(f −1 (V )). Let T be a sheaf in Y we define the inverse image presheaf f −1 T as the presheaf in X defined, f −1 T (U ) = T (U ) =

lim T (V ).

→ f (U )⊆V

Its associated sheaf is called the inverse image of T , and denoted f ∗ (T ). When Z is a topological subspace of X, we often denote by T |Z the inverse image in Z of a sheaf in X. We call this sheaf the restriction of T to Z.

A.2. Algebraic Varieties

A.2

161

Algebraic Varieties

A.2.1

Affine Varieties

Let R be a ring. An ideal of R is a subset a ⊂ R verifying that, if a, b ∈ a then the difference a − b is also in a, and if a ∈ a, b ∈ R then the product ab is also in a. An ideal a is an abelian subgroup of R. The quotient subgroup R/a inherits the ring structure of R. We have an exact sequence of abelian groups, 0 → a → R → R/a → 0. For any subset S ⊂ R there is a minimum ideal (S) containing S. Let {ai }i∈Λ a set of ideals. Then, (1) (2)

T

i ai

P

i ai

=(

S

i ai )

are ideals of R. If a, b, are ideals, then a · b = ({ab : a ∈ a, b ∈ b}), is an ideal called the product ideal. A non zero element a ∈ R, is called a divisor of zero if there exist b 6= 0 such that ab = 0. A ring without divisors of zero is called an integral domain. An ideal is called proper if it is distinct of the total. A ring without proper ideals, except (0), is a field. An ideal p ⊂ R is called prime if the quotient R/p is an integral domain. An ideal m is called maximal, if it is not strictly contained in any proper ideal,√or equivalently if R/m is a field. Any maximal ideal is prime. The radical a of an ideal a is the intersection of all prime ideals containing a. A non zero element a ∈ R is called nilpotent if there exist n with an = 0. A ring without nilpotent elements is called √ a reduced ring. The set of all nilpotent elements is called of all prime ideals of R. An the nil-radical R and it is the intersection √ ideal a is called radical ideal if a = a, or equivalently R/a is a reduced ring. A ring is called noetherian if each ideal can be described as (S) with S ⊂ R a finite set. A subset S ⊂ R is called a multiplicative system if 1 ∈ S, it does not contain the zero element, and for all s, t ∈ S, the product st belongs also to S. Let us consider the set of fractions, o na : a ∈ R, s ∈ S , s

162


up to the following equivalence relation: as ∼ bt if and only if there exist u ∈ S such that u(at − bs) = 0. This set is denoted S −1 R. There is a canonical map R → S −1 R sending a 7→ a1 . It inherits the ring structure of R, and the above morphism is a ring morphism. If R is an integral domain, then this morphism is injective. Let us consider f ∈ R a non-nilpotent element. Then S = {1, f, f 2 , . . .} is a multiplicative system. We write Rf instead of S −1 R, and we call it the localized ring in f . Let us consider a prime ideal p ⊂ R, then S = R − p is a multiplicative system. We write Rp instead of S −1 R, and we call it the localized ring in p. Rp is a local ring, it has one only maximal ideal. Let a be S any ideal of R. We consider the following multiplicative system S = R − a⊆p p, where p goes along all prime ideals containing a. Then we write Ra for S −1 R. We are going to construct the space Spec(R) associated to R. As a set, we define Spec(R) to be the set of all prime ideals of R. If a ⊂ R is any ideal we define (a)0 to be the set of all prime ideals of R containing a.

these subsets of Spec(R) verify, (1) (0)0 = Spec(R); (2) (1)0 = ∅; (3) (a · b)0 = (a)0 ∪ (b)0 ; T P (4) ( i ai )0 = i (ai )0 ;

(5) (a)0 ⊆ (b)0 if and only if

√

a⊇

√ b.

We consider in Spec(R) the topology whose closed subsets are precisely subsets of the form (a)0 with a ⊆ R. From de above properties we deduce that these subsets define a topology, called Zariski topology in Spec(R). From now on we consider Spec(R) as a topological space. Note that a point x ∈ Spec(R) is closed if and only if it corresponds to a maximal ideal. Proposition A.2.1 Spec(R) is a quasi-compact, T0 separated, topological space. ˜ ˜ Now we are going to define a sheaf ` of rings R in Spec(R). We define R(U ) to be the set of functions s : U → x∈Spec(R) Rx , such that s(x) ∈ Rx for each


163

x ∈ Spec(R) and that for each x ∈ U there is a neighborhood V of x and elements a, b ∈ R such that for each y ∈ V , b 6∈ y, we have s(y) = ab ∈ Ry . Let U ⊂ Spec(R) be an open subset. We S can define the multiplicative system SU = {a ∈ A : ∀x ∈ U (a 6∈ x)} = R − x∈U x. We denote RU for SU−1 R. ˜ ) = RU . Then, R(U Let a ∈ R. We denote Ua to the open subset Spec(R) − (a)0 , and it is called an affine open subset. This family of open subsets is a basis of the Zariski topology in the sense that each open subset is union of basic open subsets. ˜ a ) = Ra , because if b 6= an for all n, then there is a prime Moreover, R(U ideal containing b and no containing a.

Morphisms Let us consider a ring morphism ϕ : R → R′ . It means an application which commutes with sum, ϕ(a + b) = ϕ(a) + ϕ(b), with multiplication ϕ(ab) = ϕ(a)ϕ(b) and such that ϕ(1) = 1. Let a be an ideal in R′ , then ϕ−1 (a) is an ideal of R. Moreover, if a is prime, then ϕ−1 (a) is also prime. So that there is a map: ϕ∗ : Spec(R′ ) → Spec(R),

x 7→ ϕ∗ (x) = ϕ−1 (x)

which is continuous for the Zariski topology. The following examples are quite relevant: • Let a ∈ R be an ideal, and π : R → R/a be the canonical projection. Then, π ∗ : Spec(R/a) ֒→ Spec(R) is an homeomorphism of Spec(R/a) with (a)0 , the set of prime ideals containing a. • Let us consider an element a ∈ R and the localization morphism l : R → Ra . Then, l∗ : Spec(Ra ) ֒→ Spec(R) is an homeomorphism of Spec(Ra ) with Ua , the set of prime ideals not containing a.

164


Let x be a prime ideal of R. The ring Rx is a local ring, we denote by mx its only maximal ideal. Then, Rx /mx is a field that we denote by κ(x) and call it the rational field or the quotient field of x. Let f ∈ Spec(R). We denote by f (x) the class of the fraction f1 in κ(x). Then f can be seen as a function, f : Spec(R) →

a

x 7→ f (x).

κ(x),

x∈Spec(R)

Definition A.2.1 Let k be a field. A k-algebra is a ring A provided with a ring morphism i : k ֒→ A. Let A be a k-algebra and denote by X its spectrum Spec(A). Note that for all x ∈ X, Ax and κ(x) are also k-algebras. We write X(k) ⊂ X for the subset of points whose rational fields κ(x) are precisely k. An element f ∈ A defines then a function, f : X(k) → k,

x 7→ f (x).

A is said of finite type or finitely generated over k if there are elements ξ1 , . . . , ξn generating A in the sense that the morphism k[x1 , . . . , xn ] → A sending xi 7→ ξi , is surjective. Proposition A.2.2 Assume that A is of finite type. Then x ∈ X is closed if and only if k ֒→ κ(x) is an algebraic extension. Note that if k is algebraically closed, then X(k) is the subset of closed points of X.

Stalk Formula Let us consider a ring morphism ϕ : R → R′ , and a ⊂ R an ideal. We write ϕ(a) · R′ for the ideal of R′ spanned by the image of a by ϕ. Theorem A.2 (Stalk formula) Let us consider x ∈ Spec(R). The stalk (ϕ∗ )−1 (x) ⊂ Spec(R′ ) is homeomorphic to the spectrum of R′ϕ(x)·R′ /ϕ(x) · Rϕ(x)·R′ = R′ /ϕ(x) · R′

ϕ(x)·R′

= R′ ⊗R κ(x).


165

Let us note that we do two different processes in the computation of the stalk. ′ ′ First there is a process of localization: the spectrum of Rϕ(x)·R ′ = R ⊗R Rx is identified with the set of prime ideals y ⊂ R′ verifying ϕ(y) ⊆ x. Second there is a process of restriction, the spectrum of R′ /ϕ(x) · R′ = R′ ⊗R R/x is identified with the set of prime ideals y ⊂ R′ verifying ϕ(y) ⊇ x. These processes commute. When we take both together we obtain R′ ⊗R κ(x). As expected, the canonical morphism R′ → R′ ⊗R κ(x), a 7→ a ⊗ 1 induces de immersion of the stalk into Spec(R′ ).

A.2.2

Schemes

Locally Ringed Spaces Definition A.2.2 A locally ringed space is a pair (X, OX ) consisting of a topological space X, and a sheaf of rings OX such that the stalk OX,x is a local ring for each x ∈ X. A morphism of locally ringed spaces (X, OX ), (Y, OY ) is a pair (f, f ♯ ) of a continuous map f : X → Y and a morphism of sheaves f ♯ : OY → f∗ OX such that for each x ∈ X the induced morphism fx♯ : OY,f (x) → OX,x is a local morphism in the sense that (fx♯ )−1 (mx ) ⊆ mf (x) . By abuse of notation, we write X for a locally ringed space, instead of the pair (X, OX ). When we want to reference to the topological space associated to X we will write |X|. ˜ is a locally ringed space. Let R be a ring. The pair (Spec(R), R) Definition A.2.3 An affine scheme is a locally ringed space which is isomorphic to Spec(R) for some ring R. A morphism of affine schemes is a morphism of locally ringed spaces between two affine schemes. There is a one-to-one correspondence between ring morphisms and affine scheme morphisms. If ϕ : R → R′ is a ring morphism, ˜ ′ → f∗ R ˜ is the sheaf then it induces a continuous map f = ϕ∗ , and f ♯ : R ′ morphism which assigns to U ⊂ Spec(R ), the morphism f (U ) = ϕU , ϕU : R′U → Rf −1 (U ) ,

ϕ(a) a 7→ , b ϕ(b)

note that this morphism is well definedSbecause ϕ(SU ) ⊆ Sf −1 (U ) , where SU denote the multiplicative system R′ − x∈U x.

166


Reciprocally a morphism of affine schemes f : Spec(R′ ) → Spec(R) induces a morphism R → R′ , this is just the morphism f ♯ between the ring Spec(R)) of global sections. Definition A.2.4 A scheme is a locally ringed space X, such that there exist an open covering {Xi }i∈Λ (with OXi = OX |Xi ) formed by affine schemes. Let us consider a scheme X. We say that X is connected if |X| is a connected topological space. We say that X is irreducible if |X| is an irreducible topological space. We say that X is reduced if OX is a sheaf of reduced rings, if and only if the stalk OX,x is reduced for each x ∈ X. We say that X is integral if OX is a sheaf of integral rings. We say that X is locally noetherian if it can be covered by open affine subsets Xi such that OX (Xi ) is a noetherian ring; in such case can be proven that each open affine subset Xi ⊂ X is the spectrum of a noetherian ring. X is noetherian if it is locally noetherian and quasi-compact, or equivalently if can be covered by a finite number of noetherian affine subsets. If X is noetherian, then |X| is a noetherian topological space in the sense of that each descending sequence of closed subsets ends. A scheme is integral if it is reduced an irreducible (see [Ha1977] p. 82-83).

Scheme Morphisms A morphism of schemes is a morphism of locally ringed spaces between two schemes. Definition A.2.5 Let S be a scheme. A S-scheme is a scheme X together with a scheme morphism X → S. Let k be a field. A scheme morphism X → Y between two S-schemes is a S-scheme morphisms, if it commutes with the given morphisms onto S. A k-scheme is a scheme X such that OX is a sheaf of k-algebras. A scheme morphism X → Y between two kschemes if a k-scheme morphism if it induces k-algebra morphisms between their sheaves. The notion of k-scheme is the same that the notion of Spec(k) -scheme. We prefer the first notation. Given an scheme S, the S-schemes and the Sschemes morphisms define a category. This category admits direct products, the fibred product over S. If X and Y are S-schemes then there exist an unique X ×S Y such that, for all S-scheme Z: HomS (Z, X) × HomS (Z, Y ) = HomS (Z, X ×S Y ).


167

It is an extension of the tensor product. For affine k-schemes we have: Spec(A) ×k Spec(B) = Spec(A ⊗k B). In a more general setting, if X is a k-scheme covered by affine subsets Xi = Spec(Ai ) and Y is a k-scheme covered by affine subsets Yj = Spec(Bj ), then X ×k Y is covered by affine subsets Zij = Spec(Ai ⊗k Bj ). Let us consider X a S-scheme, and a morphism T → S. Then X ×S T , endowed with the natural projection onto the second factor is a T -scheme. We write XT for this scheme, and say that it is obtained from X by base extension to T . For a k-scheme X and an extension k → A, we denote by XA the extended scheme X ×k Spec(A). Definition A.2.6 A morphism f : X → Y is of finite type if there exist a covering of Y by open affine subset Yi = Spec(Ai ), such that of each i, f −1 (Yi ) can be covered by a finite number of affine subsets Xij = Spec(Bij ) being Bij finitely generated over Ai . A k-scheme X is of finite type if it can covered by a finite number of affine subsets Xi = Spec(Bi ) being each Bi a k-algebra of finite type. An open immersion is a morphism i : U ֒→ X such that: i(U ) is an open subset of X, and the structure sheaf OU is isomorphic to the restriction OX |U . A closed immersion is a morphism i : Y ֒→ X, such that i(Y ) is a closed subset of X and the restriction OX → i∗ OY is surjective. Give a closed subset Y ⊂ X there are different scheme structures realizing Y as sub-scheme of X, but there is a smallest structure, called reduced induced closed subscheme structure. In the affine case it is done as follows: let Y ⊂ Spec(R) a closed subset, and take I(Y ) = {a ∈ R : ∀x ∈ Y (a ∈ x)}. Then Spec(R/I(Y )) is the reduced induced closed subscheme structure. Note that I(Y ) is a radical ideal. Definition A.2.7 Let f : X → Y be a scheme morphism. The diagonal morphism ∆ : X → X ×Y X is the unique morphism whose composition with both projections is the identity map. We say that f is separated (or that X is a separates Y -scheme) if ∆ is a closed immersion. If ∆(X) ⊂ X ×Y X is a closed subset then f is closed (see [Ha1977] p. 96). A morphism f : Spec(A) → Spec(B) between affine schemes is always

168


separated. Just note that the diagonal projection comes from the morphism A ⊗B A → A, a ⊗ b 7→ a · b, which is surjective. For noetherian schemes the following statements hold: (a) Open and close immersions are separated. (b) Composition of separated morphisms are separated and moreover, if g ◦ f is separated, then f is separated. (c) Separated morphism are stable under base extension. (d) A morphism f : X → Y is separated if and only if Y can be covered by open subsets Yi such that f −1 (Yi ) → Yi is separated. Definition A.2.8 A morphism f : X → Y is proper, if it is separated, of finite type and universally closed. Here we say that it is universally closed if it remains closed after any base extension S → Y .

Algebraic Varieties Definition A.2.9 An algebraic variety over k in a reduced separated scheme of finite type over k. If it is proper over k, we will also say that it is complete. In particular, by Hilbert’s basis theorem, algebraic varieties are noetherian schemes. They are locally the spectra for finite generated k-algebras. For a closed point x ∈ X, we call algebraic dimension of X at x to the Krull dimension dim(OX,x ), that is the maximal length n of chains of prime ideals p0 ⊂ p1 ⊂ . . . ⊂ pn = mx. We call geometric dimension of X at x to the number dimκ(x) mx /m2x . A point x ∈ X is called a regular point if geometric and algebraic dimension coincide, if and only if OX,x is a regular ring. The dual space (mx /m2x )∗ = Tx X is canonically identified with the space of derivations Derκ(x) (OX,x , κ(x)), which is called tangent space of X at x. Definition A.2.10 An algebraic variety is smooth if each closed point is regular. Theorem A.3 Let X be an algebraic variety over k. Then X is complete if only if for every k → K extension, every valuation ring k ⊂ R ⊂ K, and


169

every morphism Spec(K) → X there exist a unique morphism Spec(R) → X such that the following diagram is commutative: Spec(K)

/ ;X ww w ww ww w w

Spec(R)

Example. The scheme Ank = Spec(k[x1 , . . . , xn ]) is an affine algebraic variety, the affine n-dimensional space over k. It is smooth. Example. The projective space P(n, k) = Proj(k[t0 , . . . , tn ]) is a smooth and complete algebraic variety (see [Ha1977], pp. 74-76). An algebraic variety over k is a projective variety if it can be embedded as a closed subset P(n, k) for some n. An algebraic variety over k is a quasi-projective variety if it can be embedded as a locally closed subset of P(n, k) for some n.

A.2.3

Functor of Points

Definition A.2.11 Let X be an k-scheme, and k ֒→ A a k-algebra. We write X(A) for the set of k-scheme homomorphisms Spec(A) → X. The functor X : A ; X(A) = Homk (Spec(A), X) of the category of k-algebras in the category of sets, is called the functor of points of X. An element x ∈ X(A) is called an A-point of X. First, note that for each field extension k ֒→ K there is a map, X(K) → X,

x 7→ x((0)),

(0) ⊂ K

following this map, X(k) is identified with the set of points of X whose rational field κ(x) is k. We call these points rational points of X. For any field extension k ⊂ K, the map X(K) → X is surjective onto the subset of points x ∈ X for whom that there exist a commutative diagram, k BB

BB BB BB !

/K = { {{ { {{ {{

κ(x)

170


and moreover, X(K) is identified with the set of K-rational points of the K-scheme XK : X × Spec(K) oo7 ooo o o o ooo

Spec(K)

/X

¯ 7→ |X|cl ⊂ |X| is surjective onto the subset If X is of finite type, then X(k) of closed points of X. Theorem A.4 There is a canonical one-to-one correspondence between the set X(K) of K-points of X and the set of rational points of the extended scheme XK . Proposition A.2.3 (Base change formula) Let X be a k-scheme, x ∈ X, and k ⊂ A a k-algebra. The stalk π −1 (x) of x by π : XA → X, is isomorphic to Spec(κ(x) ⊗k A). Proof. First, assume that X = Spec(B) is affine. Then, by stalk formula, we have π −1 (x) = Spec(A ⊗k B ⊗B ⊗κ(x)) = Spec(A ⊗k κ(x)), the homeomorphism is induced by the ring morphism A ⊗k B → A ⊗ κ(x),

a ⊗ f 7→ a ⊗ f (x).

If X is not affine, then we cover it with affine subsets Ui . If π(y) = x, and x ∈ Ui , then y ∈ Ui ×k Spec(A) and the previous argument is sufficient. 2

Specialization From now on all considered extensions of k are subfields of certain universal extension. An extension k ⊂ U is an universal extension if the following holds: for any finitely generated extension k ⊂ K with K in U and any finitely generated extension K ⊂ L, not necessarily in U, there exist a Kalgebra embedding from L to U. Universal extensions always exist (see [Ko1973]). This assumption is done in order to consider the functor of points as a set.


171

Definition A.2.12 Let us consider an extension k ⊂ K. Let us consider two k-algebras morphisms, into two different extensions of K, σ : K → L and τ : K → N . Let us consider the K-algebras K[σK] ⊂ L and K[τ K] ⊂ N . We say that σ specializes to τ , and write σ → τ if there is a commutative triangle of k-algebra morphisms, K[σK] ,

w; σ www w ww ww K GG GG GG τ GGG #

K[τ K]

we say that σ and τ are related by generic specialization if σ → τ and τ → σ, and we write σ ↔ τ . Generic specialization is a equivalence relation into the set Homk (K, •) of k-homomorphisms of K into extensions of K. In the set Homk (K, •), the notion of specialization defines a topology. We can set S¯ = {x ∈ Homk (K, •) : ∃y ∈ S(y → x)}. The quotient space Homk (K, •)/ ∼ inherits This topology. Next, we show that this correspondence gives us an homeomorphism: (Homk (K, •)/ ∼) ≃ Spec(K ⊗k K). Proposition A.2.4 Let X be a k-scheme, and let x ∈ X. Let us consider π : X ×k Spec(κ(x)) → X. Then there is a canonical continuous map, Homk (κ(x), •) → π −1 (x), witch commutes with generic specialization Homk (κ(x), •)/ ∼

∼

− →

π −1 (x),

and that induces a one-to-one correspondence between Homk (κ(x), κ(x)) and the set of κ(x)-rational points of π −1 (x). Proof. By the stalk formula π −1 (x) = Spec(κ(x) ⊗k κ(x)). Let us note that the morphism Spec(κ(x) ⊗k κ(x)) → Xκ(x) that sets the κ(x)-algebra structure of κ(x) ⊗k κ(x) is the natural inclusion in the second factor i2 : κ(x) → κ(x) ⊗k κ(x), a 7→ 1 ⊗ a. Let us consider the other natural inclusion

172


i1 : κ(x) → κ(x) ⊗k κ(x). Let us consider an extension κ(x) ⊂ L, and σ ∈ Homk (κ(x), L). This σ defines a κ(x)-algebra morphism, (σ · 1) : κ(x) ⊗k κ(x) → L,

a ⊗ b 7→ σ(a) · b.

We consider the application ψ : Hom(κ(x), •) → Spec(κ(x) ⊗k κ(x)),

σ 7→ ker(σ · 1).

In order to continue the proof we have to show: (1) σ ↔ τ if and only if ψ(σ) = ψ(τ ) (2) If ψ(τ ) ∈ {ψ(σ)}, then σ → τ . It is clear that (2) implies (1). Let us see (2). If ψ(τ ) ∈ {ψ(σ)} then ker(τ · 1) ⊂ ker(σ · 1). In such case there exist a commutative diagram as in definition A.2.12. Thus, σ → τ . Finally let us see that this map is a one-to-one correspondence of the set Homk (κ(x), κ(x)) with the set of rational points of Spec(κ(x) ⊗k κ(x)). For this, let us build an inverse for ψ onto this subset. Let x be a rational point, it is characterized by the morphism x : κ(x) ⊗k κ(x) → κ(x). Then we define σx : κ(x) → κ(x) as σx (a) = x(i1 (a)). Then, φ(σx ) = x, and we finish the proof. 2 Example. Let k ⊂ K a Galois extension. We have the spectra morphism Spec(K) → Spec(k). After base change, if becomes a morphism of Kschemes, π : Spec(K) ×k Spec(K) → Spec(K), Spec(K) consist in a unique rational point x, then π −1 (x) = Spec(K ⊗k K) which, in the case of a Galois extension, coincides with Autk (K).

B Algebraic Groups and Homogeneous Spaces

In this appendix we set out our notation and conventions in the subject of algebraic groups and homogeneous spaces. Some of this material is valid the complex analytic frame, and we use it in Chapter 2. Hence, we present before what is feasible in the complex analytic frame, and then the algebraic theory. The basic reference for the general theory is Grothendieck’s [SGA3], from who we take some theoretical results. A shorter and good introduction to algebraic groups is found in [Mum1970]. The most complete monograph, known by the author, on this subject is [Sa2001] (in Spanish), and some results are taken from there.

B.1

Complex Analytic Theory

B.1.1

Complex Analytic Lie Groups

Definition B.1.1 Complex analytic Lie Group G is a group endowed with an structure of complex analytic manifold. This structure is compatible with the group law in the sense of that the map: G × G → G,

(σ, τ ) 7→ στ −1

is complex analytic. From now on, let us consider a complex analytic Lie group G. We say that a subset H is a complex analytic subgroup of G if it is both a subgroup and ¯ a complex analytic submanifold of G. In the usual topology, the closure H

174

Appendix B. Algebraic Groups and Homogeneous Spaces

of a subgroup of G is just a Lie subgroup, and it is not in general complex ¯ we con consider the complex analytic closure, analytic. Instead of H \ ˜ = H G′ G′ complex analytic. H⊂G′

This is the smaller complex analytic subgroup containing H.

Invariant Vector Fields Each element σ ∈ G defines two biholomorphisms of G as complex analytic manifold. They are the right translation Rσ and the left translation Lσ : Rσ (τ ) = τ σ,

Lσ (τ ) = στ.

The following result is true in the general theory of groups. It is quite interesting that right translations are the symmetries of left translations and viceversa. This simple fact has amazing consequences in the theory of differential equations. Theorem B.1 Let f : G → G be a map. Then, there is an element σ such that f is the right translation Rσ if and only if f commutes with all left translations. Let us consider X(G) the space of all regular complex analytic vector fields in G. It is, in general, an infinite dimensional Lie algebra over C – with the Lie bracket of derivations –. A biholomorphism f of G transform regular vector field in regular vector fields. By abuse on notation we denote by the same symbol f the automorphism of the Lie algebra of regular vector fields. f : X(G) → X(G),

~ 7→ f (X) ~ ~ σ = f ′ (X ~ f −1 (σ) ) X f (X)

~ ∈ X(G) is called a right invariant Definition B.1.2 A regular vector field A ~ into itself. vector field if for all σ ∈ G the right translation Rσ transform A ~ A regular vector field A ∈ X(G) is called a left invariant vector field if for ~ into itself. all σ ∈ G the right translation Lσ transforms A ~ e ∈ Te G in the identity element of G there is an only For any tangent vector A ~ which takes the value A ~ e at e. Then the space right invariant vector field A of right invariants vector fields is a C-vector space of the same dimension as G. The Lie bracket of right invariant vector fields is a right invariant vector

B.1. Complex Analytic Theory

175

field. We denote by R(G) the Lie algebra of right invariants vector fields in G. The same discussion holds for left invariants vector fields. We denote by L(G) the Lie algebra of left invariants vector fields in G. Note that the inversion morphism that sends σ to σ −1 is an involution that conjugates right translations with left translations. Then, it sends right invariant vector fields to left invariant vector fields. So that: Proposition B.1.1 R(G) and L(G) are isomorphic Lie algebras.

Exponential Map ~ ∈ R(G). There is a germ of solution curve of A ~ that sends Let us consider A the origin to the identity element e ∈ G. By the composition law, it extend to a globally defined curve: σ : C → G,

t 7→ σt .

This map is a group morphism and {σt }t∈C is a monopoarametric subgroup of G. Theorem B.2 We call exponential map to the application that assigns to ~ the element σ1 of the solution curve σt of each right invariant vector field A ~ A: ~ 7→ exp(A) ~ = σ1 . exp : R(G) → G, A The exponential map is defined in the same way for left invariant vector ~ ∈ R(G) and B ~ ∈ L(G) take the same value at e, then they share fields. If A ~ = exp(B). ~ the same solution curve {σt }t∈C , and exp(A) ~ is a right invariant vector field, then the exponential of A ~ codifies the If A ~ flow of A as vector field. This flow is given by the formula: ~

ΦA : C × G → G,

~

~ · σ. ΦA : (t, σ) 7→ exp(tA)

~ is a left invariant vector The same is valid for left invariant vector fields. If B ~ field then the flow of B is: ~

ΦB : C × G → G,

~ ~ ΦB : (t, σ) 7→ σ · exp(tB)

Finally we see that right invariant vector fields are infinitesimal generators of monoparametric groups of left translations. Reciprocally left invariant vector fields are infinitesimal generators of manoparametric groups of right translations.

176


~ be a vector field in G. It is a right invariant vector Theorem B.3 Let X ~ in G the Lie bracket field if and only if for all left invariant vector field B ~ ~ B, X vanish. We have: The same is true if we change the roles of right and left in the previous statement. We have seen that the Lie algebra of symmetries of right invariant vector fields is the algebra of left invariant vector fields and viceversa: [R(G), L(G)] = 0.

B.1.2

Complex Analytic Homogeneous Spaces

Definition B.1.3 A G-space M is a complex analytic manifold M endowed with a complex analytic action of G, a

G×M − → M,

(σ, x) 7→ σ · x.

Definition B.1.4 A G-space M is a homogeneous G-space if the action of G in M is transitive. A G-space M is principal homogeneous G-space if the action of G in M is free and transitive. By a morphism of G-spaces we mean a morphism f of complex analytic manifolds which commutes with the action of G, f (σ ·x) = σ f˙(x). Whenever it does not lead to confusion we omit we write homogeneous space instead of G-homogeneous space on so on. Example. Let us consider H a closed complex analytic subgroup of G. The space of cosets, G/H = {σ · H | σ ∈ G} is an homogeneous G-space. Reciprocally, let M be an homogeneous space, and let us choose as origin a point x of M . Then, the natural map G → M , σ 7→ σ · x gives us an isomorphism of homogeneous spaces between the space of cosets G/Hx and M . Let M be a G-space. For a point x ∈ M we denote by Hx the isotropy subgroup of x, and Ox for the orbit G · x of x. The natural map, G → Ox

σ 7→ σ · x

induces an isomorphism of Ox with the homogeneous space of cosets G/Hx . A point x ∈ M is called a principal point if and only if Hx = {e} if and only

B.1. Complex Analytic Theory

177

if Ox is a principal homogeneous space. In such case we say that Ox is a principal orbit. Definition B.1.5 For a G-space M we denote by M/G the space of elements whose elements are the orbits of the action of G in M . The main problem of the invariant theory is the study of the structure of such quotient spaces. In general this space of orbits M/G can have a very complicated structure, and then it should not exist within the category of complex analytic manifolds. This situation is even more complicated in the algebraic case.

Fundamental Vector Fields ~ of the Lie algebra R(G) can be considLet M be a G-space. An element A ered as a vector field in the cartesian product G × M that acts in the first ~ is projectable by the action, component only. This vector field A a : G × M → M, ~ M , the fundamental and its projection in M is a vector field that we denote A ~ vector field in M induced by A. Proposition B.1.2 The map defined: R(G) → X(M ),

~→A ~M , A

is a Lie algebra morphisms. The kernel of this map is the Lie algebra of the kernel subgroup of the action. Thus, if the action is faithful then it is injective. The image of the map above is isomorphic to a quotient of R(G). Definition B.1.6 We denote by R(G, M ) the Lie algebra of fundamental vector fields of the action of G on M . The flow of fundamental fields is given by the exponential map in the group G: ~M ~ · x. ΦA : C × M → M, (t, x) 7→ exp(tA)

178


Basis of an Homogeneous Space From now on, let us consider an homogeneous space M . Definition B.1.7 Let S be any subset of M . We denote by HS to the isotropy subgroup of S: \ HS = Hx = {σ | σ · x = x ∀x ∈ S}. x∈S

Definition B.1.8 We call hSi space spanned by S ⊂ M to the space of invariants of HS : hSi = {x | σ · x = x ∀σ ∈ HS }. Definition B.1.9 A subset S ⊂ M is called a system of generators of M as G-space if hSi = M . S is a system of generators if and only if the isotropy HS is coincides with HM , \ \ Hx . Hx = x∈S

x∈M

In particular, if the action of M is faithful it means that S is a system of generators of M if an only if HS = {e}. Definition B.1.10 We call a basis of M to a minimal system of generators of M . We say that an homogeneous space is of finite rank if there exist a finite basis of M . The minimum cardinal of basis of M is called the rank of M. Let us consider an homogeneous space M . The group G acts in each cartesian power M r component by component, so that it is a G-space. Proposition B.1.3 Let us assume that M is a faithful homogeneous space of finite rank r. Let us consider x ¯ = (x(1) , . . . , x(r) ) ∈ M r . The following statements are equivalent: (1) {x(1) , . . . , x(r) } is a basis of M . (2) x ¯ is a principal point of M r . (3) Hx¯ = {e}.

B.2. Algebraic Groups

179

Proposition B.1.4 (Lie’s inequality) If M is a faithful homogeneous space of finite rank r then, r · dim M ≥ dim G. Proof. The dimension of M r is r · dim M . It contains principal orbits, of the same dimension as G. 2 ~1, . . . , A ~s} Remark B.1.1 Let us assume that M is faithful. Consider {A r r a basis of the Lie algebra R(G, M ) of fundamental fields in M . For each x ¯ ∈ M r , those vector fields span de tangent space to Ox¯ , ~ 1,¯x , . . . , A ~ s,¯x i = Tx¯ Ox¯ ⊂ Tx¯ M r . hA By hypothesis there is a principal point y¯ ∈ M r . The dimension of Oy¯ is s, ~ 1,¯y , . . ., A ~ s,¯y are linearly independent. Thus, the set so that A ~ 1,¯x , . . . , A ~ s,¯x are linearly independent } B0 = {¯ x ∈ Mr | A is an analytic open subset of M . We have that x ¯ is in B0 if and only if Hx¯ is a discrete subgroup of G. The function in B0 that assigns to each x ¯ the cardinal #Hx¯ of its isotropy subgroup is upper semi-continuous; thus, it reach its minimum 1, along an analytic open subset B ⊂ B0 . This B is the set of all principal points of M r , which is an analytic open subset of M r .

B.2

Algebraic Groups

From now on, let k be a field. Whenever we need it, we will assume that k is algebraically closed, and of zero characteristic. Definition B.2.1 An algebraic group over k is an algebraic variety G (in the sense of Definition A.2.9) endowed with k-algebraic morphisms (of multiplication and inversion), m : G ×k G → G,

i: G → G

verifying group axioms (G1), (G2) and (G3). Group axioms:

180


(G1) Associativity. The following diagram commutes, G ×k G ×k G

m×Id /

G ×k G

m

m

Id×m

G ×k G

/G

(G2) Identity element. There exists e ∈ G(k) such that, Id×e

/G×G KKK KKIdK e×Id KKK m K% m /G G ×k G

G KK

(G3) Inversion. The following diagram commutes,

i×Id

Id×i

/ G ×k G LL LLe LL m LL L m L/%

G LL

G ×k G

G

Theorem B.4 Any algebraic group over a field k of characteristic zero is smooth. Proof. See [Mum1970] pp. 101–102.

2

Remark B.2.1 Alternatively we can define algebraic groups as in [SGA3], as schemes whose functor of points take values in the category of groups. For an algebraic group G defined over k and any k-algebra A, the set G(A) of A-points of G is a group. Example. Let us consider GL(n, k) the spectrum of the ring k[xij , ∆]i,j=1,...,n with ∆ = |xij |−1 the inverse of the determinant. The law of composition of matrices if algebraic, so that it defines an structure of affine algebraic group in GL(n, k). For any extension k ⊂ K, GL(n, k)(K) is the group of invertible n × n matrices with coefficients in K. We call algebraic linear groups to the Zariski closed subgroups of GL(n, k). Any affine algebraic group is isomorphic to a algebraic linear group.


181

Theorem B.5 Let G be an affine algebraic group over k. Then then there is a closed embedding of groups G ⊂ GL(n, k) for some n. Definition B.2.2 An abelian variety A over k is a complete, geometrically ¯ is connected alconnected algebraic group. It means that that is A×k Spec(k) ¯ gebraic group over the algebraic closure k of k. In particular abelian varieties do not have non-constant global functions. Definition B.2.3 A quasi-abelian variety A over k is an algebraic group over k such that OA (A) = k. In particular an abelian verity is quasi abelian. Theorem B.6 Let G be an algebraic group. There is a unique subgroup X ∈ G such that, X is quasi-abelian and G/X is an affine group. Theorem B.7 (Chevalley-Barsotti-Sancho) Let G be a connected algebraic group over k, with k an algebraically closed field of characteristic zero. Then there is a unique normal affine subgroup N ⊂ G such that the quotient G/N is an abelian variety.

A remarkable fact: field automorphisms groups are not algebraic An astonishing fact is that for a field extension k ⊂ K, the group Autk (K) is not in general an algebraic group. As it as seen above Homk (K, K) is identified with the set of rational points of Spec(K ⊗k K). Then, it is expected it to be a k-algebraic group, but the multiplication law is not a k-scheme morphism, but a K-scheme morphism. Let us consider R = K ⊗k K. Let us remind that the K-algebra structure in R is given by the second inclusion i2 : i2 : K → R,

a 7→ 1 ⊗ a.

There are natural morphisms of k-algebras: e∗ : R → K, µ∗ : R ⊗K R → R,

a ⊗ b 7→ a · b,

a ⊗ b 7→ (a ⊗ 1) ⊗ (b ⊗ 1)

and, i∗ : R → R,

a ⊗ b 7→ b ⊗ a.

182


Denote by G the spectrum Spec(R). The first morphism, e∗ is by definition, a K-rational point of G. It is obvious that it corresponds to the identity element e ∈ G.

Let us consider σ, τ ∈ Autk (K). There is a one to one correspondence between k-endomorphisms of K and K-rational points of Spec(R). Let us denote σ ¯ , τ¯ to the rational points corresponding to σ and τ . They are morphisms σ ¯ , τ¯ : R → K. The pair (¯ σ , τ¯) is then a rational point (¯ σ , τ¯) : R ⊗K R → K. Thus, we have a commutative diagrams R

µ∗

/ R ⊗K R

R

(¯ σ ,¯ τ)

τ −1 ·σ

K

τ

/K

i∗

σ−1 σ

K

/R σ ¯

/K

Taking kernels in such diagrams allow us to see how the induced morphisms µ, i act over rational points that are automorphisms, µ : G ×K G → G, i : G → G,

(σ, τ ) 7→ τ −1 σ, σ 7→ σ −1 .

Then, multiplication law, and inversion are defined algebraically over G, but these morphisms are not morphisms of K-schemes, but of k-schemes. Then, Autk (K) is not in general an algebraic group over K. The exception are precisely algebraic extensions, and pro-algebraic Galois extensions, for what R is a trivial K-algebra and Autk (K) = Spec(R) is an algebraic group over K. Remark B.2.2 This previous discussion seems to be incoherent with the coring structure (see [Kov2003] Proposition 17.1) in the differential case. It is necessary to note that there is a slight difference in the definition of the tensor product R ⊗K R under Kovacic’s point of view. He takes different Kalgebras structure in each factor, by assuming (1⊗a)⊗(1⊗1) = (1⊗1)⊗(a⊗1) he takes R ⊗K R = (R, i2 ) ⊗ (R, i1 ), but in the above text we consider the same K-algebra structure in both factors.

B.2.1

Composition Law for Non-rational Points

Let G be an algebraic group over k. For any k-algebra A, the set of A-point of G, is a group G(A) (see [Sa2001] definition 1.1). In particular G(k) is the


183

group of rational points of G. When k is not algebraically closed, we say ¯ are geometric points of G, where k¯ is the algebraic that the points of G(k) closure of k. For a k-algebra morphism, A → B we have a group morphism G(A) → G(B). In particular, for A ⊂ B we have G(A) ⊂ G(B). If σ ∈ G(A) and τ ∈ G(B), then by the natural inclusions A → A⊗k B, we have σ·τ ∈ G(A⊗B). For any common extension k ⊂ A, B ⊂ T ,there is a unique morphism A ⊗k B → T , and then a morphism G(A ⊗ B) → G(T ), applying σ · τ into a T -point. We set, • For two A-points, σ and τ , we denote σ · τ for the A-point product. • For a k-point σ and a A-point τ , then σ · τ is a A-point. Lemma B.8 Assume that k is algebraically closed. Then for σ, τ ∈ G κ(σ) ⊗k κ(τ ) is an integral k algebra. Definition B.2.4 Let σ, τ ∈ G. The quotient morphism σ ∗ : OG,σ → κ(σ), defines a κ(σ) point, σ ¯, σ ¯ : Spec(κ(σ)) → G, such that σ ¯ (p0 ) = σ. Then σ ¯ · τ¯ is a (κ(σ)⊗κ(τ ))-point. Let us consider the zero ideal p0 ∈ Spec(κ(σ)⊗κ(τ )). Define, σ·τ =σ ¯ · τ¯(p0 ). Theorem B.9 G is a topological semigroup with the above definition, but not a group. Proof. Let us note that for a non-rational point σ, σ · σ −1 is a non-rational point whose rational field is the field of quotients of κ(σ) ⊗k κ(σ), different 2 of e. In fact we have e ∈ {σ · σ −1 }. Definition B.2.5 For σ ∈ G(k) we call right translation by σ, Rσ , to the k-algebraic variety automorphism in G induced by composition with σ by the right side, Rσ : G → G τ 7→ τ · σ,

and left translation by σ, Lσ , to the automorphism induced by composition with σ by the left side, Lσ : G → G

τ 7→ σ · τ.

If σ ∈ G(K) with k ⊂ K, then Rσ and Lσ are automorphisms of GK . If σ ∈ G then it induces translations of Gκ(σ) .

184

B.2.2


Lie Algebra of an Algebraic Group

Definition B.2.6 A regular vector field is a derivation A : OG → OG of the sheaf of regular functions on G. It means that for each affine open U , A|U is a derivation of the ring OG (U ). The space of regular vector fields of G is denoted X(G). For each σ ∈ G, the natural inclusion {σ} ⊂ G induces a derivation Aσ ∈ Tσ G, Aσ : OG,σ → κ(σ), it is called the value of A at σ. The Lie bracket of regular vector fields is a regular vector field, so X(G) is a Lie algebra. Definition B.2.7 Let A be a regular vector field in G, and ψ : G → G an automorphism of algebraic variety. Then, we define ψ(A) the transformed vector field ψ(A) = (ψ ♯ )−1 ◦ A ◦ ψ ♯ . OG

ψ(A)

/ OG O (ψ♯ )−1

ψ♯

OG

A

/ OG

Definition B.2.8 The Lie Algebra R(G) of G is the space of all regular vector fields A ∈ X(G) such that for all closed point σ ∈ G, Rσ (A) = A. In the same way, we define L(G) the Lie algebra of left invariant vector fields. The Lie bracket of two right invariant vector field is a right invariant vector field. The same is true for left invariant vector fields, so R(G) and L(G) are Lie sub-algebras of X(G). Theorem B.10 Assume that k is algebraically closed and of zero characteristic. Then for each tangent vector Ae ∈ Te G are unique A ∈ R(G) and B ∈ L(G) having the value Ae at e. Proof. see [Mum1970] pp. 98-99.

2

Theorem B.11 R(G) and L(G) are isomorphic Lie algebras of the same dimension. Proof. The inversion morphism i : G → G conjugates the right translation Rσ , into the left translation Lσ−1 . Then, if A is a right invariant vector field, then i(A) is a left invariant vector field. 2

B.3. Algebraic Homogeneous spaces

185

Remark B.2.3 A remarkable fact is that the isomorphism of k-vector spaces induced by the identification R(G) ≃ Te G ≃ L(G) is not a Lie algebra isomorphisms. It lacks of the inversion, which is infinitesimally seen as a change of sign. The, we have:

B.3

TeO G

/ Te G O

AO e

/ −Ae O

R(G)

/ L(G)

A

/ i(A)

Algebraic Homogeneous spaces

Let G be an algebraic group over a field k of characteristic zero. If necessary we assume that k is algebraically closed. Definition B.3.1 A G-space M is an algebraic variety over k endowed with an action of G, a G ×k M − → M, (σ, x) 7→ σ · x. Let M be a G-space. Then for each extension k ⊂ K, the group G(K) acts on the set M (K). Given a point x ∈ M its isotropy subgroup is an algebraic subgroup of G that we denote by Hx . It is defined by equation Hx · x = x. Note that it is not necessary for x to be a rational point. The intersection of the isotropy subgroups of all closed points of M is a normal algebraic subgroup HM ⊳ G. The action of G is called faithful if HM is the identity element {e}, and it is called free if for any rational point Hx = {e}. It is called transitive if for each pair of rational points x, y ∈ M there is a σ ∈ G such that σ · x = y; ide est there is only one orbit. Definition B.3.2 Let us consider the induced morphism, (a × Id) : G ×k M → M ×C M,

(σ, x) 7→ (σx, x)

then, (1) M is an homogeneous G-space if (a × Id) is surjective. (2) M is a principal homogeneous space if (a × Id) is an isomorphism.

186


Remark B.3.1 In the case of k algebraically closed definitions are simpler. An homogeneous space is an G-space with a transitive action. A principal homogeneous space is a G-space with a free and transitive action. Remark B.3.2 Assume that k is algebraically closed. For an homogeneous space M , the isotropy subgroups of closed points Hx are conjugated subgroups of G. The universal geometric quotient G/Hx exist and it is isomorphic to M . Remark B.3.3 If k is algebraically closed, any principal homogeneous space is isomorphic to G.

Chevalley’s theorem One interesting question is, given a subgroup H ⊂ G: there exists the geometric quotient space G/H as an algebraic homogeneous G-space? It is part of the algebraic invariant theory (see [Mum1970], [Sa2001]). In the case of affine groups there is always an affirmative answer due to Chevalley’s theorem. The linear group GL(V ) of linear transformation of a vector space V , acts on any tensorial space over V . Given a tensor T we call stabilizer subgroup of T to the group of linear transformations σ ∈ GL(V ) for whom there exist a scalar λ ∈ K such that σ(t) = λT. Theorem B.12 (Chevalley, see [Hu1975] p. 80) Let E be a k-vector space of finite dimension, and let H ⊂ GL(V ) be an algebraic subgroup. There exist a tensor, M ∗ V ⊗ni ⊗k V ⊗mi T ∈ i

such that H is the stabilizer of T ,

H = {σ ∈ GL(V )|hσ(T )i = hT i} From this result we obtain that for a linear algebraic group G and an algebraic subgroup H, the quotient space G/H is isomorphic to the orbit OhT i L ∗ ⊗n ⊗m i i in the projective space P ⊗k (V ) . It is an quasiprojective i V algebraic variety. Affine algebraic groups are Zariski closed subgroups of certain general linear group (Theorem B.5), so that the above result holds also for them.

B.3. Algebraic Homogeneous spaces

187

On the Existence of Quotients in the Non-linear Case In [S-P1994] (vol. IV p. 18) it is stated that arbitrary quotients exist when the base field k is algebraically closed and of characteristic zero. In [SGA3] (Séminaire 3.1 p. 315) it is stated that the category of abelian varieties over k is an abelian category: arbitrary quotients exist. This, together with Chevalley-Barsotti-Sancho theorem (Theorem B.7) should lead us to the existence of arbitrary quotients, that should be geometric and universal over an algebraically closed field of characteristic zero. We have not find a complete proof of this point in the literature. Because of that, the existence of certain quotients is included in the hypothesis of our results when necessary.

B.3.1

Fundamental Fields

Let G be an algebraic group and M an smooth scheme of finite type endowed ~ ∈ R(G) be a right invariant vector field. It with a left action of G. Let A ~ induces a vector field A ⊗ 1 in G ×k M . This vector field is projectable by the action a. ~ M ∈ X(M ) of A ~ ⊗ 1 is called the fundaDefinition B.3.3 The projection A ~ mental field in M associated to A. Proposition B.3.1 The map, R(G) → X(M ),

~ 7→ A ~M , A

is a Lie algebra morphism. It is injective if and only if the action of G in M is faithful. We denote by R(G, M ) to the Lie algebra of fundamental vector fields of the action of G on M . In particular, the Lie algebra of fundamental fields R(G, G) in G coincides with R(G).

B.3.2

Basis of Algebraic Homogeneous Spaces

We go now to the question of the rank and basis of algebraic homogeneous spaces. Throughout this text let k be an algebraically closed field of zero characteristic and M an algebraic homogeneous G-space. We consider the same definitions as we did in the analytic case. For a subset S ⊂ M , we denote HS to the isotropy subgroup of S, the intersection of the isotropy

188


subgroups of elements of S. It is an intersection of algebraic subgroups and hence it is an algebraic subgroup of M . For S ⊂ M we denote by hSi the space spanned by S the space of invariants of HS . We say that a set S of rational points of M is a system of generators if the space hSi spanned by S coincides with M . We say that a system of generators is a basis if it is minimal. The minimum between the cardinal of basis of M is called the rank of M . Remark B.3.4 Let us assume that G acts faithfully in M . Let us consider x ¯ = (x(1) , . . . , x(r) ) ∈ M r (k). Then, the following are equivalent: (1) {x(1) , . . . , x(r) } is a basis of M . (2) Hx¯ = {e}. (3) Ob is a principal homogeneous space. Theorem B.13 Any algebraic G-homogeneous space is of finite rank. Proof. Let us consider x1 in M (k), and let H1 be the isotropy subgroup of x1 . If H1 is different to HM then there exist x2 such that Hx2 is not contained in H1 . Let us define, H 2 = H 1 ∩ H x2 . We continue this process and we obtain a sequence of rational points {xi }i∈N , and a sequence of algebraic groups, H1 ⊇ H2 ⊇ . . . ⊇ Hi ⊇ . . . where, Hi =

\

H xj .

j≤i

G is an algebraic variety, hence a noetherian scheme. Therefore, any descendent chain of algebraic groups stabilizes. There is a r such that Hr =

r \

H xi = H M .

i=1

The finite set {x1 , . . . , xr } is a system of generators of M , so that it contains a basis. M is of finite rank. 2

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Index

G-space, 185 affine hull, 79 algebraic group, 179 lie algebra, 184 algebraic variety, 168 automorphic equation, 46, 99 system, 35, 40 algebraic, 95 base change formula, 170 basis, 178 Borel subgroup, 140 differential K-algebra, 68 alg. dyn. system, 92 extension, 68 strongly normal, 73 universal, 73 field, 68 ideal, 68 point, 88 ring, 67 almost-constant, 71 scheme, 81 affine, 81 almost-constant, 82 spectrum, 76 unit, 80

zero, 80 equation automorphic, 100 flag, 146 matrix Riccati, 145 projective Riccati, 146 Weierstrass, 134 flag variety, 140 functor of points, 169 fundamental solution, 105 Galois bundle, 56 cohomology, 121 correspondence, 74, 85, 114 extension, 106 group, 109 gauge transformation, 100 gauge transformation, 48 germ, 158 grassmanian, 141 homogeneous space, 185 principal, 185 isotropy subgroup, 185 Keigher ring, 69 Kolchin extension, 135 Kolchin topology, 76

198

Lie’s reduction method, 50 Lie-Vessiot system, 24 algebraic, 95 hierarchy, 42 Lie-Vessiot-Guldberg algebra, 21 liouvillian extension, 135 strict, 135 logarithmic derivative algebraic, 98 analytic, 45 Picard-Vessiot extension, 107 Pl¨ ucker coordinates, 142 embedding, 142 presheaf, 157 pretransitive, 22 Ritt algebra, 69 scheme, 166 affine, 165 morphism, 166 with derivation, 85 almost-constant, 89 sheaf, 159 special group, 50 specialization, 170 spectrum, 161 split of differential schemes, 82 of schemes with derivation, 88 splitting extension, 103 stalk, 157 stalk formula, 164 superposition law, 19 local, 25 theorem Chevalley, 186 Chevalley-Barsotti-Sancho, 181

Index

Darboux on rigid movements, 150 Drach-Kolchin, 135 global superposition, 27 Grauert, 50 Kolchin of reduction, 127 Lie’s superposition, 21 local superposition, 26 theta functions, 131 transversal symmetries, 62 universal extension, 170

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