Symplectic classification of 2-forms in dimension 4

125 Abstract of 41st Annual Iranian Mathematics Conference 12-15 September 2010, University of Urmia, Urmia-Iran

Symplectic classification of 2-forms in dimension 4 Mehdi Nadjafikhah Department of Mathematics, Iran University of Science and Technology, P. O. Box: 16765-163, Tehran, Iran. Seyed Reza Hejazi1 Department of Mathematics, Iran University of Science and Technology, P. O. Box: 16765-163, Tehran, Iran.

Abstract Classification of exterior 2-forms on 4-dimensional symplectic vector space (V, Ω) with structure 2-forms Ω is considered. This classification will process by finding orbits of action GL(V ) on the space of 2-forms on 4-dimensional symplectic vector space V which is called Jacobi planes in the sequel.

Keywords: Symplectic Geometry, differential forms, Pfaffian.

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Extended Abstracts of the 41th Iranian International Conference on Mathematics 12-15 September 2010, University of Urmia, Urmia, Iran, pp 00-00

SYMPLECTIC CLASSIFICATION OF 2-FORMS IN DIMENSION 4 MEHDI NADJAFIKHAH1 AND SEYED REZA HEJAZI2∗ Abstract. Classification of exterior 2-forms on 4-dimensional symplectic vector space (V, Ω) with structure 2-forms Ω is considered. This classification will process by finding orbits of action GL(V ) on the space of 2-forms on 4-dimensional symplectic vector space V which is called Jacobi planes in the sequel.

1. Introduction and Preliminaries Let (V, Ω) be a 4-dimensional symplectic vector space, and let ω ∈ Λ2 (V ∗ ) be a 2-form. Define a linear operator Aω : V → V by the condition Xcω = Aω XcΩ for any vector X ∈ V . For any 2-form ω ∈ Λ2 (V ∗ ) the 4-form ω 2 = ω ∧ ω is proportional to the volume form Ω2 = Ω ∧ Ω, and we define the Pfaffian Pf(ω) to be the coefficient of proportionality, i.e., ω 2 = Pf(ω)Ω2 . Definition 1.1. A 2-form ω is called effective if it is annihilated by the operator ⊥: Λi (V ∗ ) → Λi−2 (V ∗ ) by the condition ⊥ (ω) = XΩ cω. Let us give a coordinate representation for Pf(ω) and Aω . Suppose that e1 , e2 , f1 , f2 is a canonical basis in V . Then 2-forms e∗1 ∧ 2000 Mathematics Subject Classification. Primary 53Dxx; Secondary 58xx, 58A10. Key words and phrases. Symplectic Geometry, differential forms, Pfaffian. ∗ Speaker. 1

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MEHDI NADJAFIKHAH, SEYED REZA HEJAZI

e∗2 , e∗1 ∧ f1∗ − e∗2 ∧ f2∗ , e∗1 ∧ f2∗ , e∗2 ∧ f1∗ , f1∗ ∧ f2∗ , constitute a basis in the space of effective 2-forms on V . Let ω = Ee∗1 ∧ e∗2 + B(e∗1 ∧ f1∗ − e∗2 ∧ f2∗ ) + Ce∗1 ∧ f2∗ − Ae∗2 ∧ f1∗ + Df1∗ ∧ f2∗ , then ω 2 = 2(AC − B 2 − DE)e∗1 ∧ f1∗ ∧ e∗2 ∧ f2∗ = (AC − B 2 − DE)Ω2 , and Pf(ω) = AC − B 2 − DE.

2. Normal Forms In this section we consider the problem of symplectic equivalence for effective forms. Pfaffian divides the space of effective forms in to there parts corresponding to signPf(ω). We say that an effective 2-form ω 6= 0 on a 4-dimensional symplectic space is hyperbolic, if Pf(ω) < 0, elliptic, if Pf(ω) > 0, and parabolic, if Pf(ω) = 0. Theorem 2.1. Let ω be an effective 2-form on a 4-dimensional symplectic space. Then there exist a canonical basis (e1 , e2 , f1 , f2 ) in V such that p 1. ω = P f (ω)(e∗1 ∧ f2∗ − e∗2 ∧ f1∗ ) if ω is elliptic, p 2. ω = −P f (ω)(e∗1 ∧ f1∗ − e∗2 ∧ f2∗ ) if ω is hyperbolic, and 3. ω = e∗1 ∧ f2∗ if ω is parabolic. Theorem 2.2. Let ω ∈ Λ2 (V ∗ ), a Lagrangian plane L ⊂ V is an ω−plane if ω|L = 0. If ω be elliptic, then L is an ω−plane iff Aω (L) = L. If ω be hyperbolic, and V = V+ ⊕ V− be the decomposition of V p into sum of symplectic planes V± = ker(Aω ∓ −P f (ω)), then L is an ω−plane iff dim(L∩V± ) = 1 and L = L∩V+ ⊕L∩V− .If ω be parabolic and Lω = ImAω be the corresponding Lagrangian plane. Then L is an ω−plane iff dim(L ∩ Lω ) = 1. 3. Jacobi planes Q By Jacobi planes we mean 2-dimensional subspaces ⊂ Λ2 (V ∗ ) in the space of 2-forms on a 4-dimensional vector space V over R. Let us fix a volume 4-form µ ∈ Λ4 (V ∗ ). We can define a symmetric bilinear form q on Π such as: q : Π×Π → R where α∧β = q(α, β)µ. We say that

CLASSIFICATION OF 2-FORMS

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a Jacobi plane Π is elliptic, if q|Π is a non-degenerate quadratic form, hyperbolic, if q|Π is a non-degenerate sign indefinite quadratic form, parabolic, if q|Π is a degenerate non-zero quadratic form and Euler, if q|Π = 0. Proposition 3.1. If ε(Π) = sign det q and Π ∈ Λ2 (V ∗ ) and ω1 , ω2 be a basis of Π then: 1. Π is elliptic iff ε(Π) = 1 and ω1 ∧ ω2 = 0, ω1 ∧ ω1 = ω2 ∧ ω2 6= 0; 2. Π is hyperbolic iff ε(Π) = −1 and ω1 ∧ω2 = 0, ω1 ∧ω1 = −ω2 ∧ω2 6= 0; 3. Π is parabolic iff ε(Π) = 0 and ω1 ∧ ω2 = ω2 ∧ ω2 = 0 ω1 ∧ ω1 6= 0; 4. Π is Euler iff q|Π = 0, and ω1 ∧ ω1 = ω1 ∧ ω2 = ω2 ∧ ω2 = 0. Theorem 3.2. Let ω1 and ω2 be the orthogonal basis in Π. Then there is a basis e1 , e2 , f1 , f2 in V such that ω1 , ω2 take one of the following normal forms. 1. Π is elliptic then, ω1 = e∗1 ∧ f1∗ + e∗2 ∧ f2∗ , ω2 = e∗1 ∧ f2∗ − e∗2 ∧ f1∗ . 2. Π is hyperbolic then, ω1 = e∗1 ∧ f1∗ + e∗2 ∧ f2∗ , ω2 = e∗1 ∧ f1∗ − e∗2 ∧ f2∗ . 3. Π is parabolic then, ω1 = e∗1 ∧ f1∗ + e∗2 ∧ f2∗ , ω2 = e∗1 ∧ f2∗ . 4. Π is Euler then, ω1 = e∗1 ∧ f1∗ , ω2 = e∗1 ∧ f2∗ . Corollary 3.3. There are four orbits of the group GL(V) on the set of Jacobian planes. References 1. B. S. Kruglikov, Classification of Monge-Amp`ere equations with two variables., CAUSTICS’98, Warsaw. 2. B. S. Kruglikov, Symplectic and contact Lie algebras with application to the Monge-Amp`ere equations, Tr. Mat. Inst. Stehlova 221 (1998), 232-246. 3. A. Kushner, Classification of mixed type Monge-Amp`ere equations., Geometry in partial Differential Equations. (1993) pp. 173-188. 1

Department of Mathematics, Iran University of Science and Technology, P. O. Box 16765-163, Tehran, Iran. E-mail address: m [email protected] 2

Same Address E-mail address: reza [email protected]