arXiv:1506.02309v2 [math-ph] 22 Jun 2015
Deformations of non semisimple Poisson pencils of hydrodynamic type Alberto Della Vedova ∗ , Paolo Lorenzoni ∗ and Andrea Savoldi ∗∗ ∗
Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca via Roberto Cozzi 53 I-20125 Milano, Italy ∗∗ Department of Mathematical Sciences, Loughborough University Leicestershire LE11 3TU, Loughborough, United Kingdom e-mails:
[email protected] [email protected] [email protected]
Abstract We study deformations of two-component non semisimple Poisson pencils of hydrodynamic type associated with Balinskiˇı-Novikov algebras. We show that in most cases the second order deformations are parametrized by two functions of a single variable. It turns out that one function is invariant with respect to the subgroup of Miura transformations preserving the dispersionless limit and another function is related to a one-parameter family of truncated structures. In two expectional cases the second order deformations are parametrized by four functions. Among them two are invariants and two are related to a two-parameter family of truncated structures. We also study the lift of deformations of n-component semisimple structures. This example suggests that deformations of non semisimple pencils corresponding to the lifted invariant parameters are unobstructed. MSC: 37K05, 37K10, 37K25, 53D45. Keywords: bi-Hamiltonian structures, integrable hierarchies.
Contents 1 Introduction
2 1
2 Linear Poisson bivectors of hydrodynamic type 2.1 Invariant bilinear forms and bi-Hamiltonian structures 2.2 Classification results . . . . . . . . . . . . . . . . . . . . 2.3 Invariants of bi-Hamiltonian structures . . . . . . . . . 2.3.1 The cases T3, N3, N5 and N6 with κ 6= 0, −1, −2. 2.3.2 The cases N4 and N6 with κ = −2 . . . . . . . .
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3 Truncated structures
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4 Lifts of Poisson structures 4.1 Complete lift . . . . . . . . . . . . . . . . . . . . . 4.2 Lift of Poisson structures of hydrodynamic type 4.3 Lift of bivectors in the loop space . . . . . . . . . 4.4 Lift of deformations . . . . . . . . . . . . . . . . . 4.4.1 Example . . . . . . . . . . . . . . . . . . .
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A Appendix: Computations of deformations A.1 First order deformations . . . . . . . . . . . . . . . . A.1.1 T3. First order deformations . . . . . . . . . . A.1.2 N5. First order deformations . . . . . . . . . A.1.3 N3, N4 and N6. First order deformations . . A.2 Second order deformations . . . . . . . . . . . . . . A.2.1 T3. Second order deformations . . . . . . . . A.2.2 N5. Second order deformations . . . . . . . . A.2.3 N3, N4 and N6. Second order deformations
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B Appendix. Lift of Frobenius structures
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C Appendix. Lift of Hamiltonian vector fields
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References
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1 Introduction Poisson pencils of hydrodynamic type and their deformations play an important role in the modern theory of integrable PDEs. Originally the study of such structures was motivated by questions arising in the theory of Frobenius manifolds, Gromov-Witten invariants and topological field theory [10, 15]. In this setting, the deformations satisfy some additional constraints (τ -structure, Virasoro constraints) and the undeformed pencil is related to a Frobenius manifold [10]. 2
A perturbative approach to the study of these deformations was developed by Dubrovin and Zhang in [15]. In their approach, the full pencil Πij λ
=
ω2ij
X
+
k
ǫ
−λ
+
(k−l+1) Aij 2;k,l (u, ux , . . . , u(l) )∂x
l=0
k≥1
ω1ij
k+1 X
X
k
ǫ
k≥1
k+1 X
(k−l+1) Aij 1;k,l (u, ux , . . . , u(l) )∂x
l=0
!
(1.1) ,
(Aij θ;k,l are homogeneous differential polynomials of degree l) is obtained via a biHamiltonian deformation procedure from the dispersionless limit ǫ → 0: ij ij k k ω2ij − λω1ij = g2ij ∂x + bij u − λ g ∂ + b u x 1 2;k x 1;k x .
(1.2)
The pencil of metrics [10, 17] gλ = g2 − λg1 defining this limit is assumed to be semisimple, meaning that there exists a special set of coordinates, the roots (r 1 , ..., r n ) of the equation det gλ = 0, such that both metrics of the pencil gλ take diagonal form. ˜ λ of the same pencil are considered equivalent if Two deformations Πλ and Π they are related by a Miura transformation of the form X u˜i = ui + ǫk Fki (u, ux , . . . , u(k) ), (1.3) k≥1
where Fki (u, ux, . . . , u(k) ) are differential polynomials of degree k. This means that two pencils belonging to the same class are related by ˜ ij = L∗i Πkl Lj , Π k λ l λ where Lik =
X s
(−∂x )s
∂ u˜i , ∂u(k,s)
L∗i k =
X ∂ u˜i ∂s . (k,s) x ∂u s
Dubrovin, Liu and Zhang proved that the equivalence classes are labelled by n functions ci (r i ) called central invariants [25, 11]. These functions are obtained by expanding the roots λi of the equation ! X ij k det g2ij − λg1ij + = 0, Aij 2;k,0 (u) − λA1;k,0 (u) p k≥1
near λi = r i : i
i
λ =r +
∞ X
λi2k p2k ,
(1.4)
k=1
3
and selecting the coefficient of p2 . The central invariants are then defined as [12, 25]: ! i X (P ki − r i P ki )2 1 λ 1 2 1 2 = i 2 Qii2 − r i Qii1 + , i = 1, . . . , n, ci = k (r k − r i ) 3 g1ii (f ) f k6=i where f i are the diagonal components of the contravariant metric g1 in canonical coordinates and Pθij (u) = Aij θ;1,2 (u),
ij Qij θ (u) = Aθ;2,3 (u),
i, j = 1, . . . , n,
θ = 1, 2.
They can also be defined by (see [16]) h i 1 ij −1 −1 li kj ci = − i Resλ=ri Tr gλ (Qλ + (gλ )lk Pλ Pλ ) , 3f ij ij ij ij ij where Qij λ = Q2 − λQ1 and Pλ = P2 − λP1 .
In this framework the following facts should be mentioned: • Each function ci depends only on the corresponding canonical coordinate r i and it is invariant with respect to Miura transformations (1.3) [25]. • Two deformations (of the same pencil) belong to the same class of equivalence if and only if they have the same central invariants [11]. • For any choice of the dispersionless limit and of the central invariants the equivalence classes are not empty. This fact, suggested by some computations (for the scalar case see [28, 2]), has been proved only recently: by Liu and Zhang in the scalar case [27] and by Carlet, Posthuma and Shadrin in the general semisimple case [8]. The proof is based on the vanishing of certain cohomology groups introduced in [25]. • Fixed the dispersionless limit ωλ and the central invariants ci (r i ) there exists a Miura transformation (1.3) reducing the pencil to the standard form [25] Πλ = ω2 − λω1 + ǫ2 LieX(c1 ,..,cn ) ω1 + ǫ4 Π4 + ǫ6 Π6 + ... = ω2 − λω1 + ǫ2 LieY(c1 ,..,cn ) ω2 + ǫ4 Π4 + ǫ6 Π6 + ...
where the polynomial vector fields X(c1 ,...,cn) and Y(c1 ,...,cn) can be written as difference of two Hamiltonian vector fields X(c1 ,...,cn) = ω2 δH − ω1 δK, 4
Y(c1 ,...,cn) = ω2 δH ′ − ω1 δK ′
with non polynomial hamiltonian densities: H[r] =
n Z X
i
c (r
i
)rxi logrxi
dx,
i=1
n Z X ci (r i ) i ′ rx logrxi dx, H [r] = i r i=1
K[r] =
n Z X
i=1 n Z X
K ′ [r] =
r i ci (r i )rxi logrxi dx. (1.5) ci (r i )rxi logrxi dx. (1.6)
i=1
• The coefficients Fk (u, ux, . . . , u(k) ) of the Miura transformation (1.3) are assumed to depend polynomially on the derivatives of ui . Removing this assumption the classification problem becomes "trivial": all deformations turn out to be equivalent to their dispersionless limit. This remarkable property of the deformations was discovered in [11] and it is called quasitriviality. For instance, it is easy to check that the canonical quasi-Miura transformation generated by the Hamiltonian H defined in the formula (1.5) reduces the pencil ij ij 4 Πij λ to the form ω2 − λω1 + O(ǫ ). In the present paper we start the study of the non semisimple case. Whereas the semisimple case is fairly understood, the non semisimple case is widely open. Beside computational difficulties, the lack of canonical coordinates, or at least of a normal form theorem for non semisimple pencils, makes very hard a unified approach to the problem. For this reason in this paper we try and get some information on the general case focusing on two special subcases where computations are feasible: The deformations of Poisson pencils related to two-dimensional Balinskiˇı-Novikov algebras [6] and the associated invariant bilinear forms. These are two component Poisson pencils that can be reduced to the form k ij ω2ij − λω1ij = g ij ∂x + bij k ux − λη ∂x ij where g ij depends linearly on the variable (u1 , ..., un ) and the coefficients bij k and η are constant. Special deformations associated with second and third order cocycles of Balinskiˇı-Novikov algebras naturally arise in the study of multi-component generalizations of the Camassa-Holm equation [34]. We will consider deformations of two component non degenerate structures related to Balinskiˇı-Novikov algebras, that is the cases T3, N3, N4 (for η 11 = 0), N5 and N6 (for κ 6= −1) of the Bai-Meng’s list [3] (which is recalled afterwards in Section 2, Table 2). The cases N1 and N4 with η 11 6= 0 are semisimple and then they are covered by Dubrovin-Liu-Zhang theory. The non semisimple structures we focus on are summarized on the next table, where we also write down the corresponding affinor L = gη −1 .
5
Table 1: Pair of metrics of bi-Hamiltonian structures. Type (T3) (N5) (N3,N4,N6)
Linear metrics g ! 0 −u1 −u1 0
! 0 u1 u1 2(u1 + u2 ) 0 (1 + κ)u1 1 (1 + κ)u 2u2
!
Constant metrics η ! 0 η 12 η 12 η 22 ! 0 η 12 η 12 η 22 ! 0 η 12 η 12 η 22
Affinors L 1
− ηu12
η 22 u1 (η 12 )2
0 1
− ηu12
u1 η 12 2(u1 +u2 ) η 22 u1 − (η 12 )2 η 12 2u2 η 12
(1+κ)u1 η 12 22 1 u − (1+κ)η (η 12 )2
! 0 u1 η 12
!
0 (1+κ)u1 η 12
!
We prove that in the cases T3, N3 (corresponding to κ = 1), N5 and N6 with κ 6= 0, −1, −2 the deformations are quasi-trivial and can be reduced to the form Πλ = ω2 − λω1 + ǫ2 LieX(F1 ,F2 ) ω2 + O(ǫ3 ) with X(F1 ,F2 ) = ω1 δH − ω2 δK where Z X H[u] = hij uix log ujx dx,
K[u] =
i,j
Z X i,j
fij uix log ujx dx,
and the functions hij and fij are uniquely determined by two arbitrary functions F1 , F2 . Moreover both functions F1 and F2 depend only on the eigenvalue of the affinor L. The cases N4 (corresponding to κ = 0) and N6 with κ = −2 are more involved and the functions labelling non Miura equivalent deformations are 4 (still depending on the eigenvalue of the affinor L). In all cases one half of the arbitrary functions parametrizing the deformations (one in the two-parameter case, two in the four-parameter case) is related to a family of truncated structures and one half is invariant with respect to the Miura transformations that preserve the dispersionless limit. The invariant functions are related to the first coefficients of the expansion (1.4) (in the second case also the odd powers of p appear in this expansion): the coefficients of p2 in the case of the algebras T3, N3, N5 and N6 with κ 6= 0, −1, −2 and the coefficients of p and p2 in the case of the algebras N4 and N6 with κ = −2. Moreover our computations suggest that in the exceptional cases generic deformations are not quasi-trivial. This fact is rather unexepcted and deserves a deeper investigation. The lift of deformations of semisimple structures. These are obtained using an infinite dimensional version of the complete lift introduced by Yano and Kobayashi in [35]. Whereas elementary, this case is important for it provides examples of full deformations of non semisimple structures depending on functional parameters. By 6
construction all deformations of a n-component semisimple structure can be lifted to deformations of a 2n-component non semisimple structure. This means that the deformations of the lifted Poisson pencils contain n functional parameters at least. This example suggests that also in the non semisimple case the deformations are unobstructed.
2 Linear Poisson bivectors of hydrodynamic type Let us introduce Poisson bivector of hydrodynamic type on the loop space L(M). The tangent space to L(M) at a loop γ : S 1 → M is naturally identified with the space Γ(S 1 , γ ∗ T M) of vector fields along γ. On the other hand (a subspace of) the cotangent space to L(M) at γ is identified with the space Γ(S 1 , γ ∗ T ∗ M) of covector fields along γ, and the pairing between a tangent vector X and a covector ξ is just R ξ(X) dx. S1 Let g be a pseudo-metric on M with Levi-Civita connection ∇. For any covector ξ ∈ Γ(S 1 , γ ∗ T ∗ M), let Xξ ∈ Γ(S 1 , γ ∗ T M) be the pointwise metric dual of ξ. Given two covectors ξ, η ∈ Γ(S 1 , γ ∗ T ∗ M), letting Z P (ξ, η) = ξ(∇γ˙ Xη ) dx S1
defines a bivector on L(M). As shown by Dubrovin and Novikov, P is a Poisson structure on L(M) if and only if ∇ is flat [13]. In local coordinates ui on M and x on S 1 the Poisson tensor P is represented by a differential operator of the form P ij = g ij (u)∂x − g il Γjlk (u)ukx ,
(2.1)
where Γjlk are the Christoffel symbols correponding to g. Dubrovin-Novikov operators naturally appear in the study of Hamiltonian quasilinear systems of PDEs uit = Vji (u)ujx ,
i = 1, ..., n,
and their dispersive Hamiltonian deformations i (u)ujxukx + O(ǫ2 ). uit = Vji (u)ujx + ǫ Aij (u)ujxx + Bjk
In this paper we will study linear Hamiltonian operators. As proved by Balinskiˇı and Novikov in [6] these operators have the form ji k ij k P ij = (bij k + bk )u ∂x + bk ux ,
7
where the numbers bij k are the structure constants of an algebra B satisfying the following properties a · (b · c) = b · (a · c),
(a · b) · c − a · (b · c) = (a · c) · b − a · (c · b). We refer to them as Balinskiˇı-Novikov algebras, even if in the literature they are often called Novikov algebras (following [33]). A first approach to the study of such algebras was made by Zelmanov [38]. In low dimensions the problem of classification was addressed by Bai and Meng [3, 5] and recently by Burde and de Graaf [7], resulting in a complete description of one-, two- and three-dimensional Balinskiˇı-Novikov algebras. Unfortunately, a full classification of these structures of dimension four and higher is far from being complete.
2.1 Invariant bilinear forms and bi-Hamiltonian structures Given a Balinskiˇı-Novikov algebra B, as observed in [34], any invariant bilinear symmetric form on it give rise to a bi-Hamiltonian structure in a canonical way. For convenience of the reader let us briefly recall how they are defined. Let e1 , . . . , en be a basis of B, and let bij k be the corresponding structure constants. A bilinear form η : B × B → F is called invariant if and only if η(ei · ej , ek ) = η(ei , ek · ej ). Bai and Meng classified these invariant bilinear forms on two- and three-dimensional Balinskiˇı-Novikov algebras in [3, 4]. For future reference we recall the two-dimensional classification in the following table. Table 2: Two-dimensional Balinskiˇı-Novikov algebras and invariant bilinear forms. Type (T1) (T2) (T3) (N1)
Characteristic matrix ei · ej ! 0 0 0 0 ! e2 0 0 0 ! 0 0 −e1 0 ! e1 0 0 e2
Linear Poisson structure ! 0 0 0 0 ! 2u2 ∂x + u2x 0 0 0 0 −u1 ∂x −u1 ∂x − u1x 0
!
2u1 ∂x + u1x 0 2 0 2u ∂x + u2x
8
!
Invariant bilinear forms ! η 11 η 12 η 21 η 22 ! η 11 η 12 η 12 0 ! 0 η 12 η 12 η 22 ! η 11 0 0 η 22
(N2) (N3) (N4) (N5)
(N6)
e1 0 0 0
!
2u1 ∂x + u1x 0 0 0
!
! e1 e2 e2 0 ! 0 e1 0 e2
2u1 ∂x + u1x 2u2 ∂x + u2x 2u2 ∂x + u2x 0 ! 0 u1 ∂x + u1x u1 ∂x 2u2 ∂x + u2x
! 0 e1 0 e1 + e2 ! 0 e1 κe1 e2 κ 6= 0, 1
0 u1 ∂x + u1x 1 1 u ∂x 2(u + u2 )∂x + u2x + u1x
!
!
0 (1 + κ)u1 ∂x + u1x 1 1 (1 + κ)u ∂x + κux 2u2 ∂x + u2x
!
η 11 0 0 η 22
!
η 11 η 12 η 12 0
!
η 11 η 12 η 21 η 22
!
0 η 12 η 12 η 22
!
0 η 12 η 12 η 22
!
Remark. Notice that the case N4 with η 11 6= 0 is semisimple. For this reason we will consider only the case η 11 = 0. The cases N3 and N4 can be considered as subcases of N6, if we remove the constraints κ 6= 0, 1. Indeed, for κ = 0 we easily get N4 (with η 11 = 0) while N3 is equivalent to the case κ = 1, up to swapping the local coordinates u1 , u2 . According to [3], this distinction is due to different algebraic properties: the cases N3 and N4 are characterized by the associativity of the algebra, while this is not the case of N6 with κ 6= 0, 1. However, for our purposes, we do not need to distinguish these cases. Let us point out that adding the constraint η 21 = η 12 in T1 and N4, the bilinear invariant forms associated with two-dimensional Balinskiˇı-Novikov algebra become symmetric. As observed by Strachan and Szablikowski in [34] the associated Hamiltonian operator η ij ∂x is compatible with the linear Hamiltonian operator defining the Balinskiˇı-Novikov algebra. Remark. A pair of compatible flat metrics defines a (2 + 1)-Poisson structure of hydrodynamic type under some additional conditions. Among the structures coming from two component Balinskiˇı-Novikov algebras, such additional conditions are satisfied just by N6 with κ = −2 [14, 31, 32, 18].
2.2 Classification results In this section we provide a classification of second order deformations of Poisson pencils coming from Balinskiˇı-Novikov algebras. By definition, a k-th deformation of a Poisson pencil of hydrodynamic type ˜ λ, Π ˜ λ ] = O(ǫk+1 ). Here where Π ˜ ij denotes (1.2) is a deformation (1.1) such that [Π λ
9
the distribution ˜ ij = ω2ij + Π
X
k
ǫ
k≥1
−λ ω1ij +
k+1 X l=0
X
ǫk
k≥1
(k−l+1) Aij (x − y) 2;k,l (u, ux , . . . , u(l) )δ k+1 X l=0
!
(k−l+1) Aij (x − y) , 1;k,l (u, ux , . . . , u(l) )δ
and the Schouten bracket is defined as follows [15]: ˜ λ, Π ˜ λ ]ijk (x, y, z) = [Π ˜ ij (x, y) ˜ ki ˜ jk ∂Π ˜ lk (x, z) + 2 ∂ Πλ (z, x) ∂ s Π ˜ lj (z, y) + 2 ∂ Πλ (y, z) ∂ s Π ˜ li (y, x), 2 λl ∂xs Π λ ∂u(s) (x) ∂ul(s) (z) z λ ∂ul(s) (y) y λ We have to distinguish two cases: 1. The cases T3, N3, N5 and N6 with κ 6= 0, −1, −2 where second order deformed structures depend on two functions. 2. The remaining cases N4 (which corresponds to κ = 0) and N6 with κ = −2, namely ! ! 0 η 12 0 ±u1 g1 = , g2 = , η 12 η 22 ±u1 2u2 where second order deformed structures depend on four functions. Theorem 1. In the cases T3, N3, N5 and N6 with κ 6= 0, −1, −2, second order deformations can be reduced by a Miura transformation to the form Πλ = ω2 − λω1 + ǫ2 LieX(F1 ,F2 ) ω2 + O(ǫ3 ) with X(F1 ,F2 ) = ω1 δH − ω2 δK where Z X H[u] = hij uix log ujx dx,
K[u] =
i,j
Z X i,j
kij uix log ujx dx,
and the functions hij and kij are uniquely determined in terms of two arbitrary functions F1 , F2 depending only on the eigenvalue of the affinor L = g2 g1−1 . Calling K = (kij ) and H = (hij ), we have K = LT H, where LT means the transpose of L, and H is given respectively for each case by • T3: h12 = h22 = 0 and 12 2
12 2
− η22 u1
h11 =
e
η
u
3η 12
η 12 u2 + η 22 u1 22 1 ′ η u F2 + F2 − F1 , u1 10
− η22 u1
h21 = −
e
η
3
u
F2 .
• N5: h12 = h22 = 0 and p 2η 12 (u1 + u2 ) − η 22 u1 F2′ F1 (2η 12 − η 22 )F2 p + 12 , + h11 = 12 12 12 1 2 22 1 3η 2η 6η 2η (u + u ) − η u 1 h21 = p F2 . 3 2η 12 (u1 + u2 ) − η 22 u1
• N3, N6 (κ 6= 0, −1, −2): h12 = h22 = 0 and (2η 12 u2 − (κ + 1)η 22 u1 ) h11 = 3(κ + 1)2 η 12 F1 + 12 , η κ(κ + 2) h21
κ+1 2
F2′
η 22 (2η 12 u2 − (κ + 1)η 22 u1 ) − 6η 12
(2η 12 u2 − (κ + 1)η 22 u1 ) = 3(κ + 1)
k−1 2
κ−1 2
F2 .
Here Fi = Fi (u1 ), i = 1, 2. In the case N4, namely g2 =
0 η 12 η 12 η 22
!
g1 =
,
! 0 u1 , u1 2u2
the second order deformations can be reduced by a Miura transformation to the form Πλ = ω2 − λω1 + ǫ2 LieX ω2 + O(ǫ3 ) where X i = X1i u1xx + X2i (u1x )2 + X3i u1x u2x + X4i (u2x )2 + X5i u2xx , with X11 = 0, X21 = θF1 , X31 = ∂1 (θF2 ) X41 = ∂2 (θF2 ), X51 = θF2 , X12 = 0, X22 = θF3 , 1 ∂1 F2 2 X3 = ∂1 θ 2 F4 − 12 , η 11
F2
X42 X52
1 ∂1 F2 = ∂2 θ 2 F4 − 12 , η 1 ∂1 F2 = θ 2 F4 − 12 . η
In the above formulas Fi are 4 arbitrary functions of u1 and θ = (η 22 u1 − 2η 12 u2 )−1 . In the case N6 with κ = −2, namely g1 =
0 η 12 η 12 η 22
!
g2 =
,
0 −u1 −u1 2u2
!
,
the second order deformations can be reduced by a Miura transformation to the form Πλ = ω2 − λω1 + ǫ2 LieX ω2 + O(ǫ3 )
(2.2)
where X i = X1i u1xx + X2i (u1x )2 + X3i u1x u2x + X4i (u2x )2 + X5i u2xx , with X11 = 0, X21
∂1 (θ2 F2 ) = 2η θ θ F4 − η 12
22
3 2
5
X31 = 2η 12 θ 2 F4 − ∂1 (θ3 F2 ),
+ θF1 ,
X41 = −4η 12 θ4 F2 , X51 = θ3 F2 ,
X12 = 0, X22 = F3 , 3
X32 = ∂1 (θ 2 F4 ) −
∂12 (θ2 F2 ) , η 12 3
X42 = 4∂1 (θ3 F2 ) + ∂2 (θ 2 F4 ), 3 ∂1 (θ2 F2 ) 2 2 X5 = θ F4 − . η 12 In the above formulas Fi are 4 arbitrary functions of u1 and θ = (2η 12 u2 + η 22 u1 )−1 . Due to its technical nature, we postpone the proof to Appendix A. Corollary 2. In the cases T3, N3, N5 and N6 with κ 6= 0, −1, −2, all second order deformations are quasi-trivial. 12
Proof: By construction the canonical quasi-Miura transformation generated by H[u] reduces the pencil to its dispersionless limit up to terms of order O(ǫ4 ). Remark. General Miura transformations have the form X ui → u˜i = f i (u) + ǫk Fki (u, ux , . . . , u(k)). k≥1
i
∂f where det ∂u j 6= 0. In this paper we are interested in Miura transformations preserving the disperionless limit and for this reason we consider the subgroup X ui → u˜i = ui + ǫk Fki (u, ux, . . . , u(k) ). k≥1
Indeed, the only diffeomorphism preserving both metrics of the pencil is the identity.
2.3 Invariants of bi-Hamiltonian structures As already mentioned in the Introduction, the central invariants for deformations of semisimple Poisson pencils of hydrodynamic type (1.1) are related to the roots of the equation ! X ij k ij ij ij = 0. A2;k,0(u) − λA1;k,0 (u) p det g2 − λg1 + k≥1
Expanding these roots near λi = r i one obtains a series: λi = r i +
∞ X
λik pk ,
(2.3)
k=1
whose coefficients are invariants (up to permutations) with respect to Miura transformations as shown by Dubrovin, Liu and Zhang in [12]. Due to the skew-symmetry of the pencil, the sum and product of the roots contain only even powers of p. In the semisimple case also the expansions (2.3) of the roots contain only even powers of p, while in the non semisimple case, in general also odd powers are allowed. For instance, in the case of deformations of non semisimple pencils associated with Balinskiˇı-Novikov algebras one obtains the expansions 1
1
λ =u +
∞ X k=1
λ1k pk ,
2
1
λ =u +
∞ X k=1
13
λ2k pk .
(2.4)
where, due to skew-symmetry: λ12k+1 + λ22k+1 = 0,
λ12k − λ22k = 0.
(2.5)
Thus it is natural to divide Poisson pencils associated with Balinskiˇı-Novikov algebras in two classes: those admitting as invariants λ11 = −λ21 and λ12 = λ22 (and eventually higher order coefficients of the expansions (2.4)) and those admitting as invariants only λ12 − λ22 (and eventually higher order coefficients of the expansions (2.4)). 2.3.1 The cases T3, N3, N5 and N6 with κ 6= 0, −1, −2. In the cases T3, N3, N5 and N6 with κ 6= 0, −1, −2, the expansions of λi do not contain the linear term in p and the coefficients of the quadratic terms λ12 = λ22 are related to the functional parameter F2 . Theorem 3. Let ωλ = ω2 − λω1 bi-Hamiltonian structure corresponding to one of the Balinskiˇı-Novikov algebras T3, N3, N5 and N6 with κ 6= 0, −1, −2 and the associated symmetric bilinear invariant form η. Let us consider a bi-Hamiltonian structures Πλ of the form (1.1) with leading term ωλij . Then the coefficients λ12 and λ22 of the expansion (2.3) coincide and they are related to the functional parameter F2 by the formulas: • T3: λi2 =
u2 u1 − ηη12 22 u1 e F2 (u1 ). 12 η
• N5: λi2 = −
η 12
p
u1 F2 (u1) . 2η 12 (u1 + u2 ) − η 22 u1
• N3, N6 with κ 6= 0, −1, −2:
λi2
(κ + 1)u1 (2η 12 u2 − (κ + 1)η 22 u1 ) =− η 12
κ−1 2
F2 (u1 ).
Proof: We are going to prove this statement in the case T3 with η 22 6= 0. In this case the dispersionless limit is given by ! ! ! 12 1 0 η 0 0 0 −u ω1ij = ∂x + . ∂x , ω2ij = 12 22 1 η η −u 0 −u1x 0 If we write the pencil in the standard form Πij λ
=
ωλij
+
2 X k=1
k
ǫ
k+1 X l=0
(k−l+1) ij Aij (u, . . . , u ) − λA (u, . . . , u ) ∂x + O(ǫ3 ) (l) (l) 2;k,l 1;k,l 14
the first two terms of the expansion (1.4) are λi2 = 0 12 12 (P212 )2 η 22 Q11 u1 Q12 1 2 1 + P1 P2 12 i + + , λ2 = 12 Q2 + η u1 2η 12 η 12
(2.6) (2.7)
where Pθij (u) = Aij θ;1,2 (u),
ij Qij θ (u) = Aθ;2,3 (u),
i, j = 1, . . . , n,
θ = 1, 2.
We know from general theory that these coefficients are invariant up to permutations. The condition λ12n = λ22n implies that are genuine invariants. Using this the proof is a straightforward computation: substituting the relations ! η 12 u2 1 1 − η22 u1 F2 (u ) 0 0 0 ue , P1 = P2 = Q1 = , Q2 = η 12 u2 − 0 0 u1e η22 u1 F2 (u1 ) ∗
in the formula (2.7) we get the result. Remaining cases can be proved following the same procedure. Remark. The invariant λi2 can be also written as 1 λi2 = − Resλ=λˆ Tr(gλ−1Λλ ) 2 ˆ is the eigenvalue of the affinor L = g2 g −1 and Λij = Qij + 1 (g −1)lk P li P kj . where λ 1 λ λ λ λ 2 λ 2.3.2 The cases N4 and N6 with κ = −2
In the remaining cases the expansion of λi contains also the linear term in p and the invariants λ11 = −λ21 and λ12 = λ22 are related to the functional parameters F2 and F4 respectively. Theorem 4. Let ωλ = ω2 − λω1 bi-Hamiltonian structure corresponding to one of the Balinskiˇı-Novikov algebras N4 and N6 with κ = −2 and the associated symmetric bilinear invariant form η. Let us consider a bi-Hamiltonian structures Πλ of the form (1.1) with leading term ωλij . Then, the invariants (λi1 )2 and λi2 are related to the functional parameters F2 and F4 through the formulas: • N4: (λi1 )2 λi2
2u1F2 , = (η 12 )3 u1 F4 ∂1 (u1F2 ) p − . = (η 12 )2 η 12 −2η 12 u2 + η 22 u1 15
• N6, κ = −2: 2u1 F2 , (η 12 )3 (2η 12 u2 + η 22 u1 )2 u1 F4 (2η 12 u2 − η 22 u1 )F2 + u1 F2′ = 12 12 2 − . η (2η u + η 22 u1 )3/2 (η 12 )2 (2η 12 u2 + η 22 u1 )3
(λi1 )2 = λi2
Proof: We outline the proof in the case N4 (corresponding to κ = 0). In this case, the standard form of the pencil is ˜ ij = ω ij + ǫ2 Θij + O(ǫ3 ) = ω ij + ǫ2 Θij ∂ 3 + Θij ∂ 2 + Θij ∂x + Θij + O(ǫ3 ), Π λ λ λ (3) x (2) x (1) (0) where
ωλij =
! 0 u1 ∂x + u1 2u2
0 u1x 0 u2x
!
0 η 12 − λ 12 22 η η
!
∂x .
and
Θ(3) = u1 F ′
2
η12
u1 F2′ η12
2u1 F2 2η12 u2 −η22 u1
−√
u1 F 4 12 −2η u2 +η22 u1
+
2u2 F2 2η12 u2 −η22 u1
2u2 F2 u1 F 4 + 12 2η u2 −η22 u1 −2η12 u2 +η22 u1 , 2 ′ 2 4u F2 4u F4 √ − η12 −2η12 u2 +η22 u1
−√
From the general theory and from relations (2.5) we know that (λi1 )2 and λi2 are invariants. Using the invariance the proof is a straightforward computation. The case N6 with κ = −2 can be treated in a similar way. Remark. The function Θ12 (3) can be also written as Θ12 (3) = −
η 12 Resλ=λˆ Tr(gλ−1Λλ ) 2
ˆ is the eigenvalue of the affinor L = g2 g −1 and Λij = Qij + 1 (g −1)lk P li P kj . where λ 1 λ λ λ λ 2 λ
3 Truncated structures In Theorems 3, 4 we proved the invariant nature of some functional parameters appearing in deformations. In this section we prove that the remaining parameters are related to truncated structures. These are Poisson pencils of the form (1.1) depending polynomially on the parameter ǫ (that is the sum in (1.1) contains finitely many terms). We show that setting to zero the invariant parameters we obtain deformations that are Miura equivalent to truncated pencils up to the order three. 16
More precisely we prove that in the cases T3, N3, N5 and N6 with κ 6= 0, −1, −2 the additional parameter provides a one-parameter family of truncated structures, while in the cases N4 and N6 with κ = −2 the two additional parameters provide a two-parameter family of truncated structures. Theorem 5. In the cases T3, N3, N5 and N6 with κ 6= 0, −1, −2, the second order deformations with F2 = 0 can be reduced by a Miura transformation to the form Πλ = ωλ + ǫ2 Θ + O(ǫ3 ) where ! ! ! 0 0 0 0 0 0 Θ= ∂x3 + ∂x2 + ∂x , (3.1) 0 2f 0 3fx 0 fxx with f = f (u1 ). Moreover the truncated pencil ωλ + ǫ2 Θ is a Poisson pencil. Proof: The form (3.1) can be easily obtained from the results of Theorem 1 rescaling the function F1 . In particular, we have to set • F1 (u1 ) =
f (u1 ) , for T3, u1
• F1 (u1 ) = −
η 12 f (u1 ) , for N5, u1
• F1 (u1 ) = −
η 12 κf (u1 ) , for N3, N6 with κ 6= 0, −1, −2. (1 + κ)u1
To prove that ωλ + ǫ2 Θ is a Poisson pencil, we have to show that 1 [Θ, Θ]ijk (x, y, z) = 2 ∂Θki (z, x) s lj ∂Θjk (y, z) s li ∂Θij (x, y) s lk ∂ Θ (x, z) + ∂ Θ (z, y) + ∂ Θ (y, x) = 0. ∂ul(s) (x) x ∂ul(s) (z) z ∂ul(s) (y) y Taking into account that Θ11 = Θ12 = Θ21 = 0 and
∂Θ22 ∂u2(s)
= 0, we obtain the result.
Theorem 6. In the case N6 with κ = −2 the second order deformations with F2 = F4 = 0 can be reduced by a Miura transformation to the form Πλ = ωλ + ǫ2 Θ + O(ǫ3 ) where ! ! ! ! 0 0 0 0 0 0 0 0 Θ= ∂x3 + ∂x2 + ∂x + , (3.2) 0 2f 0 3fx 0 fxx + 2g 0 gx with f = f (u1) and g = (h(u1 )u1x )x + h(u1 )u1xx . Moreover the truncated pencil ωλ + ǫ2 Θ is a Poisson pencil. 17
Proof: Here we prove only the first part of the theorem. The second part can be obtained as above by straightforward computation. By Theorem 1 we have Πλ = ω2 − λω1 + ǫ2 LieX ω2 + O(ǫ3 ), where the component of the vector field X are given by X 1 = θF1 (u1x )2 ,
X 2 = F3 (u1x )2 ,
with θ = (2η 12 u2 + η 22 u1)−1 . The Miura transformation ui → exp(−ǫY )ui ,
i = 1, 2,
generated by the vector field Y of components Y 1 = −η 12 Ru1xx − η 12 ∂1 R(u1x )2 − η 12 ∂2 Ru1x u2x ,
Y 2 = −η 22 Ru1xx − η 22 ∂1 R(u1x )2 + (η 12 ∂1 R − η 22 ∂2 R)u1x u2x + η 12 ∂2 R(u2x )2 + η 12 Ru2xx , with R = where ˜1
X
˜2 X
u1 F 1 , 2η12 (2η12 u2 +η22 u1 )
reduces the pencil to the form ω2 −λω1 +ǫ2 LieX˜ ω2 +O(ǫ3 ),
1 ′ θu F1 θu1 F1 u1xx 2 12 2 22 1 − − θ (η u + η u )F1 (u1x )2 + θ2 η 12 u1 F1 u1x u2x = − 2 2 1 ′ 22 1 1 θη u F1 uxx θu1 F1 u2xx θu F1 2 12 2 22 2 = − + + + θ (η u + η u )F1 u1x u2x 2η 12 2 2 22 1 ′ θη u F1 2 22 2 + θ η u F1 − F3 (u1x )2 − θ2 η 12 u1 F1 (u2x )2 . − 2η 12 12
To conclude it is easy to check that LieX˜ ω2 coincides with (3.2) (F1 = − 2ηu1 f and F3 = − uh1 ). Theorem 7. In the case N4 with F2 = F4 = 0 the second order deformations can be reduced by a Miura transformation to the form Πλ = ωλ + ǫ2 Θ + O(ǫ3 ) where ! ! ! ! 12 11 12 11 12 0 0 0 q q q q q 2 Θ= ∂x3 + ∂x2 + 121 122 ∂x + 021 022 , (3.3) 0 q322 −q212 q222 q1 q1 q0 q0 with q322 = 2f, q212 = 4θη 12 f u1x , q222 = 3f ′ u1x , 18
q111 = −8(θη 12 )2 f (u1x )2 ,
q112 = (2θη 12 f ′ − 2θ2 η 12 η 22 f + 2θ2 h)(u1x )2 ,
q121 = (−6θη 12 f ′ − 10θ2 η 12 η 22 f + 2θ2 h)(u1x )2 + 16(θη 12 )2 f u1x u2x − 8θη 12 f u1xx ,
q122 = (f ′′ + 2θ(η 12 )−1 h′ + 6θ2 (η 12 )−1 η 22 h)(u1x )2 − 8θ2 hu1x u2x + (f ′ + 4θ(η 12 )−1 h)u1xx , q011 = − 4(θη 12 )2 f ′ + 8θ3 (η 12 )2 η 22 f (u1x )3 + 16(θη 12 )3 f (u1x )2 u2x − 8(θη 12 )2 f u1x u1xx ,
q012 = (2θ2 h′ + 4θ3 η 22 h)(u1x )3 − 8θ3 η 12 h(u1x )2 u2x + 4θ2 hu1x u1xx , q021 = (−2θη 12 f ′′ − 8θ2 η 12 η 22 f ′ − 12θ3 η 12 (η 22 )2 f )(u1x )3
+(12(θη 12 )2 f ′ + 40θ3 (η 12 )2 η 22 f )(u1x )2 u2x + (−8θη 12 f ′ − 16θ2 η 12 η 22 f )u1x u1xx −32(θη 12 )3 f u1x (u2x )2 + 8(θη 12 )2 f u1x u2xx + 16(θη 12 )2 f u1xx u2x − 4θη 12 f u1xxx,
q022 = (θ(η 12 )−1 h′′ + 4θ2 (η 12 )−1 η 22 h′ + 6θ3 (η 12 )−1 (η 22 )2 h)(u1x )3
+(−6θ2 h − 20θ3 η 22 h)(u1x )2 u2x + (4θ(η 12 )−1 h′ + 8θ2 (η 12 )−1 η 22 h)u1x u1xx +16θ3 η 12 hu1x (u2x )2 − 2θ2 hu1x u2xx − 4θ2 hu1xx u2x + θ(η 12 )−1 hu1xxx ,
where f = f (u1 ), h = h(u1 ) and θ = (2η 12 u2 − η 22 u1 )−1 . Moreover the truncated pencil ωλ + ǫ2 Θ is a Poisson pencil. Proof: By Theorem 1 we have Πλ = ω2 − λω1 + ǫ2 LieX ω2 + O(ǫ3 ), where the components of the vector field X are given by X 1 = −θF1 (u1x )2 ,
X 2 = −θF3 (u1x )2 ,
with θ = (2η 12 u2 − η 22 u1 )−1 . The Miura transformation ui → exp(−ǫY )ui ,
i = 1, 2,
generated by the vector field Y of components Y 1 = −η 12 Ru1xx − η 12 ∂1 R(u1x )2 − η 12 ∂2 Ru1x u2x ,
Y 2 = −η 22 Ru1xx − η 22 ∂1 R(u1x )2 + (η 12 ∂1 R − η 22 ∂2 R)u1x u2x + η 12 ∂2 R(u2x )2 + η 12 Ru2xx , 1
u F1 with R = − 2η12 (2η12 , reduces the pencil to the form u2 −η22 u1 )
ω2 − λω1 + ǫ2 LieX˜ ω2 + O(ǫ3 ), where X
1
X2
1 ′ θu F1 θu1 F1 u1xx 2 12 2 22 2 + − θ (η u − η u )F1 (u1x )2 − θ2 η 12 u1 F1 u1x u2x , = 2 2 1 ′ 22 1 1 θu F1 θη u F1 uxx θu1 F1 u2xx 2 12 2 22 2 − − + θ (η u + η u )F1 u1x u2x = 2η 12 2 2 19
+
θη 22 u1 F1′ 2 22 2 + θ η u F1 − θF3 (u1x )2 + θ2 η 12 u1 F1 (u2x )2 , 12 2η
To conclude the first part of the theorem we observe that it is easy to check that 12 LieX˜ ω2 = Θ (F1 = 2ηu1 f and F3 = − η12hu1 ). The second part is a cumbersome computation. Remark. Truncated Poisson pencils of the form Πij λ
=
ωλij
+ǫ
2 X
(Aij 2;1,l
l=0
−
(2−l) λAij 1;1,l )∂x
3 X ij (3−l) +ǫ (Aij 2;2,l − λA1;2,l )∂x 2
(3.4)
l=0
where ωλ is a Poisson pencil of hydrodynamic type associated with a BalinskiˇıNovikov algebra appear in [34]. In this case the coefficients ij ij ij Aij 2;1,0 , A1;1,0 , A2;2,0 , A1;2,0
are related with second and third order cocycles of the Balinskiˇı-Novikov algebra. In order to reduce deformations of the form (3.4) to the canonical form Πλ = ωλ + ǫ2 Θ + O(ǫ3 ) one has to peform a Miura transformation producing (in general) infinitely many terms in the right hand side of (3.4). For this reason (in general) Strachan-Szablikowski truncated pencils correspond in our framework to non truncated pencils.
4 Lifts of Poisson structures Given a differentiable manifold M, there is a natural way for lifting tensor fields and affine connections from M to its tangent bundle T M, viewed as a manifold itself. Such a lift is named complete lift and has been extensively studied by Yano and Kobayashi [35, 36, 37]. In this section we apply this construction to Poisson tensors defined on a suitable loop space.
4.1 Complete lift Let us recall the definition and some properties of complete lift, referring to original papers mentioned above for more details. Given local coordinates u1 , . . . , un on M, let u1 , . . . , un , v 1 , . . . , v n be the induced bundle coordinates on T M so that any tangent vector on M has the form v i ∂u∂ i . The complete lift of a function f , a one form α = αi dui , and a vector field X = X i ∂u∂ i is defined respectively by ∂f fˆ = v j j , ∂u
α ˆ=
∂αi v j j dui ∂u
i
+ αi dv , 20
i i ∂ j ∂X ∂ ˆ X =X +v . ∂ui ∂uj ∂v i
(4.1)
ˆ and a comIt follows readily from these local expressions that α(X) lifts to α( ˆ X), ˆ Yˆ ]. mutator [X, Y ] lifts to [X, Lifted vector fields (resp. one-forms) span the tangent (resp. cotangent) space of T M at any point which does not belong to the zero section {v = 0}. As a ˆ of any given tensor field K just by consequence, one can define the complete lift K imposing that any contraction with a vector field X or a one-form α on M lifts to ˆ with X ˆ or α. the contraction of K ˆ Then one check that exterior derivative and Lie derivative are invariant with respect to the complete lift, meaning that dξ lifts to dξˆ ˆ for any differential form ξ and that a Lie derivative LX K lifts to L ˆ K. X
It may be useful to have at hand explicit expressions for some special classes of tensors. In particular, the complete lift of a bilinear form g = gij dui ⊗ duj turns out to be gˆ = v k
∂gij i du ⊗ duj + gij dui ⊗ dv j + gij dv i ⊗ duj , ∂uk
(4.2)
and a trilinear form T = Tijk dui ⊗ duj ⊗ duk lifts to ∂Tijk i Tˆ = v h du ⊗duj ⊗duk +Tijk dui ⊗duj ⊗dv k +Tijk dui ⊗dv j ⊗duk +Tijk dv i ⊗duj ⊗duk . ∂uh Moreover, an endomorphism of the tangent bundle A = Aij ∂u∂ i ⊗ duj lifts to ∂Aij ∂ ∂ ∂ Aˆ = Aij i ⊗ duj + v k k i ⊗ duj + Aij i ⊗ dv j , ∂u ∂u ∂v ∂v
(4.3)
and the lift of a bilinear product on vector fields · = cijk ∂u∂ i ⊗ duj ⊗ duk is ˆ· =
cijk
∂cijk ∂ ∂ j k h ⊗ du ⊗ du + v ⊗ duj ⊗ duk ∂ui ∂uh ∂v i ∂ ∂ + cijk i ⊗ duj ⊗ duk + cijk i ⊗ dv j ⊗ duk . (4.4) ∂v ∂v
Finally, any bivector P = P ij ∂u∂ i ⊗
∂ ∂uj
lifts to
∂ ∂ ∂ ∂P ij ∂ ∂ ∂ ⊗ . Pˆ = P ij i ⊗ j + P ij i ⊗ j + v k k ∂u ∂v ∂v ∂u ∂u ∂v i ∂v j
(4.5)
Let now ∇ ∂u∂ k = Γijk ∂u∂ i ⊗ duj be an affine connection on M. Its complete lift ˆ is an affine connection on T M defined by requiring that for all vector fields X ∇ ˆ X. ˆ Using that ∂ k and ul ∂ k lift to ∂ k and on M the endomorphism ∇X lifts to ∇ ∂u ∂u ∂u ul ∂u∂ k + v l ∂v∂k respectively, one can check that ∂Γi ˆ ∂ = Γijk ∂ ⊗ duj + v h jk ∂ ⊗ duj + Γijk ∂ ⊗ dv j , ∇ ∂uk ∂ui ∂uh ∂v i ∂v i 21
(4.6)
ˆ ∂ = Γijk ∂ ⊗ duj . ∇ (4.7) ∂v k ∂v i Readily from definition one deduces that for any tensor field K on M the complete ˆ K. ˆ In particular, any flat tensor (∇K = 0) lifts to a flat tensor lift of ∇K equals ∇ ˆK ˆ = 0). Moreover it holds the following [35, Proposition 7.1]: (∇
ˆ are the complete lift of the torsion and Proposition 8. The torsion and the curvature of ∇ the curvature of ∇.
Remark. Since the lift it is well defined for tensors and connections we can apply it to the geometric structures defining a Frobenius manifolds. As a result one obtain a lifted Frobenius structure. We discuss this construction in more detail in the Appendix B.
4.2 Lift of Poisson structures of hydrodynamic type The class of structures that can be lifted to the tangent bundle by means of complete lift includes symplectic forms and more generally Poisson tensors. The latter has been studied in some detail by Mitric and Vaisman [30]. Since the Schouten bracket is defined in terms of Lie derivative, if follows that it is invariant by complete lift as well. As a consequence, the complete lift of a bi-Hamiltonian structure Pλ = P + λQ, where λ ∈ R and P, Q are Poisson tensors on M satisfying [P, Q] = 0, ˆ is a bi-Hamiltonian structure Pˆλ = Pˆ + λQ. Recall that, in local coordinates ui on M and x on S 1 the Poisson tensor P at γ = u(x) is represented by ∂u∂ i ⊗ P ij ∂u∂ j where k P ij = g ij ∂x + bij k ux ,
i, j = 1, ..., n.
(4.8)
ih j Here g ij is the inverse of the matrix gij which represents g locally, and bij k = −g Γhk , being Γjhk the Christoffel symbols of g. It is clear that P can be lifted to L(T M) defining Pˆ as γ Pˆ αβ = gˆαβ ∂x + ˆbαβ α, β = 1, ..., 2n, γ ux , where gˆ is the lift of the contravariant metric, ˆbαβ are the contravariant Christoffel γ
symbols of the lifted Levi-Civita connection and we set un+i = v i . Indeed one has ˆ is the Levi-Civita connection of the lifted metric gˆ. But this only to check that ∇ ˆg = 0 follows by uniqueness of Levi-Civita connection together with the fact that ∇ˆ ˆ is torsion free by Proposition 8 and by torsion-freeness of for ∇g = 0, and that ∇ ∇. Therefore gˆ defines a Poisson structure of hydrodynamic type Pˆ on L(T M). Remark. It is easy to check that the lift Pˆ is uniquely defined by the requirement (the analogous property in the finite dimensional case has been observed in [30]) Z hv, {ξ, η}P i dx (4.9) {Hξ , Hη }Pˆ = S1
22
R where Hξ = S 1 hξ, vi dx and {·, ·}P is the Poisson bracket on 1-forms [19, 29] defined by g [1]: # " ∂(η) ∂(ξ) j j (4.10) {ξ, η}j = g kl ∂xs+1 (η)l k − ∂xs+1 (ξ)l k . ∂u(s) ∂u(s) Proposition 9. In local coordinates ui , v i on T M one has ∂ ∂ ∂ k k ∂ Pˆ = i ⊗ (g ij ∂x + bij + i ⊗ (g ij ∂x + bij k ux ) k ux ) j ∂v ∂u ∂u ∂v j ! ij ∂b ∂ ∂ ji ij k h k k u + b v + i ⊗ v h (bij x x h + bh )∂x + v k ∂v ∂uh ∂v j
(4.11)
Proof: Thanks to (4.8) we have to determine the coefficients g ij and bij k for the lifted metric j gˆ. To this end, let W be the metric dual of the coordinate one-form duj on M. This means that W j is the unique vector field on M such that g(W j , ·) = duj , and clearly one has W j = g ij
∂ . ∂ui
(4.12)
Moreover, well known properties of Christoffel symbols yield ∇W j = bij k
∂ ⊗ duk . ∂ui
(4.13)
Therefore one can write P = W j ⊗ ∂x
∂ ∂ + ∇γ˙ W j ⊗ j , j ∂u ∂u
(4.14)
wehere γ˙ = ukx ∂u∂ k . Let U j and V j be the metric dual of duj and dv j with respect to the lifted metric gˆ on T M. One can readily check by (4.2) that U j = g ij
∂ . ∂v i
(4.15)
ˆ j, On the other hand, by (4.1) the lift of duj turns out to be dv j . Therefore V j = W so that V j = g ij
∂ ji ∂ + v k (bij , k + bk ) i ∂u ∂v i
(4.16)
where we used the identity ∂g ij ji = bij k + bk . k ∂u
(4.17) 23
ˆ j =∇ ˆW ˆ j , whence by definition of lifted connection and equations In particular ∇V (4.13), (4.3) it follows ij ∂ ij ∂ k h ∂bk ∂ j ˆ ⊗ duk + bij ⊗ dv k . ∇V = bk i ⊗ du + v k h i i ∂u ∂u ∂v ∂v
(4.18)
On the other hand, by (4.7) one calculates ij ˆ j = ∂g ∂ ⊗ duk + g ij Γh ∂ ⊗ duk , ∇U ki ∂uk ∂v i ∂v h
(4.19)
whence, thanks to the identity (4.17), one concludes ˆ j = bij ∂ ⊗ duk . ∇U k ∂v i
(4.20)
The statement then follows by simple calculations from equations (4.15), (4.16), (4.18), (4.20) and the identity ∂ ˆ γ˙ U j ⊗ ∂ + V j ⊗ ∂x ∂ + ∇γ˙ V j ⊗ ∂ , Pˆ = U j ⊗ ∂x j + ∇ ∂u ∂uj ∂v j ∂v j
(4.21)
where γ˙ = ukx ∂u∂ k + vxk ∂v∂k for any loop γ = (u(x), v(x)) in T M.
4.3 Lift of bivectors in the loop space In matrix notation the lift (4.11) takes the form ! 0 P ij P , Pˆ = k ∂P ij P ij k,t v(t) ∂uk
(4.22)
(t)
whence it is clear that one can lift to L(T M) any given Poisson structure (nonnecessarily of hydrodynamic type) on the loop space L(M). The proof of this fact is contained in the book [23] in the framework of linearization of Hamiltonian objects a.k.a. formal or universal linearization (see for instance [24, 21]) or tangent covering (see for instance [22]). We provide here a different direct proof which rests just on the Schouten bracket formula given in [15]. Theorem 10. Suppose that ij Px,y
=
Pkij (x
− y, u, ux, . . . , uk+1) =
k+1 X
(k+1−m) Aij (x − y). m (u, ux , . . . , uk+1 )δ
k+1 X
ij Bm (u, ux , . . . , uk+1)δ (k+1−m) (x − y).
m=0
and Qij x,y
=
Qij k (x
− y, u, ux, . . . , uk+1) =
m=0
24
have vanishing Schouten bracket [P, Q]ijk x,y,z
ki ij ∂Qij ∂Pz,x ∂Px,y x,y s lk s lk ∂x Qx,z + l ∂x Px,z + l ∂zs Qlj = z,y + l ∂u(s) (x) ∂u(s) (x) ∂u(s) (z)
+
jk ∂Py,z ∂Qjk ∂Qki y,z z,x s lj s li ∂ P + ∂ Q + ∂ s P li = 0, z z,y y y,x l l l ∂u(s) (z) ∂u(s) (y) ∂u(s) (y) y y,x
then also the lifted structures Pˆ =
0 P ij P k ∂P ij P ij k,t v(t) ∂uk
(t)
!
ˆ= Q
,
0 Qij P k ∂Qij Qij k,t v(t) ∂uk
(t)
have vanishing Schouten bracket.
!
Proof: Throughout in this proof un+i will denote v i for all i = 1, . . . , n. Moreover we fix the convention that latin indices i, j, k run from 1 through n, and greek indices α, β, γ run from 1 through 2n. By straightforward computation we obtain • For α = i, β = j, γ = k: ˆ αβγ [Pˆ , Q] x,y,z =
ki ij ˆ ij ∂Q ∂ Pˆz,x ∂ Pˆx,y x,y s ˆ λk s ˆ λk ˆ λj + ∂ Q + ∂ P + ∂sQ ∂uλ(s) (x) x x,z ∂uλ(s) (x) x x,z ∂uλ(s) (z) z z,y jk ˆ jk ˆ ki ∂Q ∂ Pˆy,z ∂Q y,z z,x s ˆ λj s ˆ λi λi ∂z Pz,y + λ ∂y Qy,x + λ ∂ys Pˆy,x = 0, + λ ∂u(s) (z) ∂u(s) (y) ∂u(s) (y)
ij ˆ ij = Pˆ ki = Q ˆ ki = Pˆ jk = Q ˆ jk = 0. since Pˆx,y =Q x,y z,x z,x y,z y,z
• For α = n + i, β = j, γ = k: ˆ αβγ = [Pˆ , Q] x,y,z
n+i,j k,n+i ˆ n+i,j ∂ Pˆx,y ∂Q ∂ Pˆz,x x,y s ˆ λk s ˆ λk ˆ λj + ∂ Q + ∂ P + ∂sQ ∂uλ(s) (x) x x,z ∂uλ(s) (x) x x,z ∂uλ(s) (z) z z,y
+
jk k,n+i ˆ jk ˆ z,x ∂ Pˆy,z ∂Q ∂Q y,z s ˆ λj s ˆ λ,n+i ∂ P + ∂ Q + ∂ s Pˆ λ,n+i = z z,y y y,x λ λ λ ∂u(s) (z) ∂u(s) (y) ∂u(s) (y) y y,x ij ∂Px,y
n+l ∂u(s) (x)
+
∂xs Qlk x,z +
∂Qki z,x n+l ∂u(s) (z)
∂Qij x,y n+l ∂u(s) (x)
lj ∂zs Pz,y +
lk ∂xs Px,z +
ki ∂Pz,x
∂zs Qlj z,y n+l ∂u(s) (z)
+
jk ˆ jk ∂Q ∂ Pˆy,z y,z s li ∂ Q + ∂ s P li = 0, ∂ul(s) (y) y y,x ∂ul(s) (y) y y,x
jk ˆ jk = 0 and P ij , Qij , P ki , Qki do not depend on coordinates since Pˆy,z =Q y,z x,y x,y x,y x,y on the fibers. Similarly one can prove the vanishing of the Schouten bracket for α = i, β = n + j, γ = k and α = i, β = j, γ = n + k.
25
• For α = n + i, β = n + j, γ = k: ˆ αβγ [Pˆ , Q] x,y,z =
n+i,n+j k,n+i ˆ n+i,n+j ∂ Pˆx,y ∂Q ∂ Pˆz,x x,y s ˆ λk s ˆ λk ˆ λ,n+j + ∂ Q + ∂ P + ∂sQ ∂uλ(s) (x) x x,z ∂uλ(s) (x) x x,z ∂uλ(s) (z) z z,y
+
n+j,k ˆ n+j,k ˆ k,n+i ∂ Pˆy,z ∂Q ∂Q y,z z,x s ˆ λ,n+j s ˆ λ,n+i ∂ P + ∂ Q + ∂ s Pˆ λ,n+i = z z,y y y,x λ λ λ ∂u(s) (z) ∂u(s) (y) ∂u(s) (y) y y,x
n+i,n+j ∂ Pˆx,y n+l ∂u(s) (x)
+
∂xs Qlk x,z
ˆ n+i,n+j ∂Q x,y
+
n+l ∂u(s) (x)
lk ∂xs Px,z
ki ∂Pz,x ∂zs Qlj + l z,y + ∂u(s) (z)
jk ∂Qki ∂Py,z ∂Qjk z,x y,z s lj s li ∂ P + ∂ Q + ∂ s P li . z z,y y y,x l l l ∂u(s) (z) ∂u(s) (y) ∂u(s) (y) y y,x
Using the identities n+i,n+j ∂ Pˆx,y
ij ∂Px,y = , n+l ∂ul(s) (x) ∂u(s) (x)
ˆ n+i,n+j ∂Q x,y
∂Qij x,y = , n+l l ∂u(s) (x) ∂u(s) (x)
(4.23)
we finally get ˆ n+i,n+j,k = [P, Q]ijk = 0. [Pˆ , Q] x,y,z x,y,z Similarly one can prove the vanishing of the Schouten bracket for α = i, β = n + j, γ = n + k and α = n + i, β = j, γ = n + k. • For α = n + i, β = n + j, γ = n + k: ˆ αβγ = [Pˆ , Q] x,y,z
n+k,n+i n+i,n+j ˆ n+i,n+j ∂Q ∂ Pˆz,x ∂ Pˆx,y x,y s ˆ λ,n+k s ˆ λ,n+k ˆ λ,n+j + ∂ Q + ∂ P + ∂sQ ∂uλ(s) (x) x x,z ∂uλ(s) (x) x x,z ∂uλ(s) (z) z z,y
n+j,n+k ˆ n+k,n+i ˆ n+j,n+k ∂Q ∂ Pˆy,z ∂Q z,x y,z s ˆ λ,n+j s ˆ λ,n+i + ∂z Pz,y + ∂y Qy,x + ∂ s Pˆ λ,n+i = λ λ ∂u(s) (z) ∂u(s) (y) ∂uλ(s) (y) y y,x n+i,n+j n+k,n+i ˆ n+i,n+j ∂ Pˆx,y ∂Q ∂ Pˆz,x x,y s lk s lk ∂ Q + ∂ P + ∂ s Qlj + ∂ul(s) (x) x x,z ∂ul(s) (x) x x,z ∂ul(s) (z) z z,y
+
n+j,n+k ˆ n+k,n+i ˆ n+j,n+k ∂Q ∂ Pˆy,z ∂Q z,x y,z s lj s li ∂ P + ∂ Q + ∂ s P li + z z,y y y,x l l ∂u(s) (z) ∂u(s) (y) ∂ul(s) (y) y y,x
n+i,n+j ∂ Pˆx,y n+l ∂u(s) (x)
+
ˆ n+l,n+k ∂xs Q x,z
ˆ n+k,n+i ∂Q z,x n+l ∂u(s) (z)
+
n+l,n+j ∂zs Pˆz,y
ˆ n+i,n+j ∂Q x,y n+l ∂u(s) (x)
+
n+l,n+k ∂xs Pˆx,z
n+j,n+k ∂ Pˆy,z n+l ∂u(s) (y)
+
n+l,n+i ˆ y,x ∂ys Q
n+k,n+i ∂ Pˆz,x n+l ∂u(s) (z)
+
ˆ n+l,n+j + ∂zs Q z,y
ˆ n+j,n+k ∂Q y,z n+l ∂u(s) (y)
n+l,n+i ∂ys Pˆy,x
Using the identities (4.23) and the fact that the operator ∂x and the operator P n+k ∂ k,t u(t) ∂uk (x) commute, as it is immediate to check using the identity (t)
∂x
∂
∂uk(t) (x)
=
∂
∂x ∂uk(t) (x) 26
−
∂
∂uk(t−1) (x)
,
we obtain n+i,n+j,n+k ˆ x,y,z [Pˆ , Q] =
X k,t
n+k u(t) (x)
∂ ∂uk(t) (x)
n+k + u(t) (y)
∂ ∂uk(t) (y)
n+k + u(t) (z)
∂ ∂uk(t) (z)
!
[P, Q]ijk x,y,z = 0
since [P, Q]ijk x,y,z = 0 by hypothesis.
Remark. Notice that the lift of bivectors (4.22) is obtained from (4.5) just replacing P j ∂ P j ∂ j v ∂uj with j,k v(k) ∂uj . The lift of general tensor fields can be defined in ex(k)
actly the same way. For instance the lift of functionals, one forms and vector fields can be defined as Z X j ∂αi X j ∂X i ∂ δF ˆ = Xi ∂ + X α ˆ= v(k) j δui +αi δv i , v(k) j . Fˆ = v j j dx, i i δu ∂u ∂v ∂u ∂u (k) (k) j,k j,k
ˆ of higher order tensor fields K can be As in the finite dimensional case the lift K defined requiring that any contraction with a vector field X or a one-form α on ˆ with X ˆ or α the loop space lifts to the contraction of K ˆ . As a consequence of this general rule the lift of a Hamiltonian vector field coincides with the Hamiltonian vector field obtained lifting the Poisson bivector and the Hamiltonian functional: ˆ In the Appendix C we check this fact. Finally we point out that [ P δH = Pˆ δ H. the linearization of Hamiltonian objects mentioned above is nothing but the YanoKobayashi complete lift in the infinite-dimensional setting.
4.4 Lift of deformations We have seen in the introduction that deformations of n-component semisimple Poisson pencils of hydrodynamic type depend on n arbitrary functions of a single variable. Applying the previous construction to this case we get a n-parameter family of deformations of the lifted Poisson pencil of hydrodynamic type. Due to obvious identity 2 detˆ π ij = ± detπ ij any invariant coefficient comes with double multiplicity. This example suggests that deformations of non semisimple structures corresponding to those invariant parameters are unobstructed.
27
4.4.1 Example In the scalar case all second order deformations are given by [28] Πλ = 2u∂x + ux − λ∂x + ǫ2 (2s∂x3 + 3sx ∂x2 + sxx ∂x ) + O(ǫ3 ),
(4.24)
where c is a constant and s(u) is an arbitrary function of u. Applying the lift we obtain a one-parameter family of deformations of a 2-component Poisson pencil of hydrodynamic type. Here we want to show this lift is equivalent, up to Miura transformations, to the case N3 (that is, N6 with κ = 1) with F1 (u1 ) = η 22 = 0. Let us consider second order deformations of N3 obtained in Theorem 1, and set η 22 = 0 (otherwise g1 would not be the lift of the scalar constant metric η = 1), η 12 = 1, F1 (u1 ) = 0 and 1) . F2 (u1) = − f (u u1 The Miura transformation ui → exp(−ǫY )ui ,
i = 1, 2,
generated by the vector field Y of components Y1 =
f ′′ f′ 1 uxx + (u1x )2 , 3 3
Y2 =−
f ′′ 1 2 f ′ 2 u u − uxx , 3 x x 3
reduces the pencil to the form ! 0 Πλ ˆλ = P Π , ∂Πλ Πλ t v(t) ∂u(t)
where Πλ coincides with (4.24) setting u1 = u and f (u1 ) = s(u).
A Appendix: Computations of deformations In this appendix we give a sketch of the proof of Theorem 1, providing the computations of deformations in detail. First of all we observe that the pencil Πij λ can be always reduced to the form Πλ = ωλ + ǫQ1 + ǫ2 Q2 + ǫ3 Q3 + ...
(A.1)
by a suitable Miura transformation. The proof is due to Getzler and it is based on the study of Poisson-Lichnerowicz cohomology groups [20] (an alternative proof can be found in [9, 15, 26]) : H j (L(Rn ), ω) :=
ker{dω : Λjloc → Λj+1 loc } j−1 im{dω : Λloc → Λjloc }
28
for Poisson bivector of hydrodynamic type ω. The differential dω is defined as dω := [ω, · ] where the square bracket is the Schouten bracket. Getzler also proved the triviality of cohomology for any positive integer j (in particular the triviality of deformations is related to the vanishing of the second cohomology group).
A.1 First order deformations The pencil (A.1) is a deformation of ωλ if it satisfies the Jacobi identity for every λ, that is [Q, Q] = [ω1 , Q] = 0. where Q = ω2 + ǫQ1 + ǫ2 Q2 + ǫ3 Q3 + .... This implies in particular [ω2 , Q1 ] = [ω1 , Q1 ] = 0. In other words Q1 is a cocycle for both the differentials dω1 and dω2 . Using the triviality of H 1 (L(Rn ), ω) and H 2(L(Rn ), ω) we obtain Q1 = dω2 X = LieX ω2 for a suitable vector field of degree 1 X i = X1i (u1 , u2 )u1x + X2i (u1 , u2 )u2x ,
i = 1, 2,
satisfying dω1 dω2 X = 0. It is not difficult to prove that among the solutions of the above equation those corresponding to trivial deformations have the form X = ω1 δH + ω2 δK, where the hamiltonian denisties are differential polynonials of degree 0, namely H = R R h(u1 , u2 ) dx and K = k(u1 , u2) dx. It turns out that in our case all first order defomations are trivial. All details below, case by case. A.1.1 T3. First order deformations Let us point out that in this case the vanishing of the coefficient η 22 implies that the affinor Lij assumes diagonal form, while for η 22 6= 0 it corresponds to one 2 × 2 Jordan block case (as well as all other cases we are dealing with). Recall that we are assuming η 12 6= 0. The vector field X solution of dω1 dω2 X = 0 is given in components by Z η 22 u1 2 2 η 22 1 1 1 1 1 1 1 2 ∂1 X1 − 12 ∂1 X1 du2 + F, X1 = X1 , X2 = X2 , X1 = 12 ∂1 (X1 u ) + η η 29
X22
=
X11
η 22 + 12 X21 + u1 (∂2 X11 − ∂1 X21 ) , η
where F = F (u1 ). The components Y i of the vector field Y = ω1 δH + ω2 δK are given by Y i = Y1i u1x + Y2i u2x , where Y11 = η 12 ∂1 ∂2 H − u1 ∂1 ∂2 K, Y21 = η 12 ∂22 H − u1 ∂22 K, Y12 = ∂1 (η 12 ∂1 H + η 22 ∂2 H − u1 ∂1 K), Y22 = ∂2 (η 12 ∂1 H + η 22 ∂2 H − u1 ∂1 K), Choosing H and K such that Xi1 = Yi1 for i = 1, 2, one can easily see that X12 = Y12 + F,
X22 = Y22 .
Finally, the function F can be removed using the vector field Y such that H = 0 and K such that −∂1 (u1 ∂1 K) = F . Thus, first order deformations are trivial. A.1.2 N5. First order deformations Here η 12 6= 0. Solving dω1 dω2 X = 0 for deg(X) = 1 we get X21 = ∂1 F, X12 = ∂2 F, Z η 22 ∂2 F + η 12 ∂1 F − η 12 X22 2 2 du2 + G, X1 = ∂1 X2 + 2η 12 (u1 + u2 ) − η 22 u1
X22 = X22 ,
where F = F (u1, u2 ) and G = G(u1 ). The components Y i of the vector field Y = ω1 δH + ω2 δK are given by Y11 = ∂1 (η 12 ∂2 H + u1 ∂2 K), Y21 = ∂2 (η 12 ∂2 H + u1 ∂2 K), Y12 = η 12 ∂12 H + η 22 ∂1 ∂2 H + u1 ∂12 K + 2(u1 + u2 )∂1 ∂2 K + ∂2 K, Y22 = η 12 ∂1 ∂2 H + η 22 ∂22 H + u1 ∂1 ∂2 K + 2(u1 + u2 )∂22 K + ∂2 K. Choosing H and K such that F = η 12 ∂2 H + u1 ∂2 K, X22 = Y22 , we obtain X11 = X21 = X22 = 0,
X12 = G.
Taking H = 0 and K such that ∂2 K = 0 and u1 ∂12 K = G, we can also remove G. Thus, deformations of degree 1 are trivial. A.1.3 N3, N4 and N6. First order deformations This case is more involved. Let us assume κ 6= −1, otherwise the metric g2 would be degenerate. Here η 12 6= 0. 30
Imposing dω1 dω2 X = 0 for deg(X) = 1 we get X11 = ∂1 G + R, X21 = ∂2 G, X12 = ∂1 F, X22 = ∂2 F, Z κ κ κ 2 κ η 22 ∂2 G + η 12 ∂1 G − η 12 ∂2 F θ−1− 2 du2 + θ 2 S, R=θ
where F = F (u1 , u2), G = G(u1 , u2), S = S(u1 ) and θ = 2η 12 u2 − (1 + κ)η 22 u1 . The components Y i of the vector field Y = ω1 δH + ω2 δK are given by Y11 = ∂1 (η 12 ∂2 H + (1 + κ)u1 ∂2 K) − κ∂2 K,
Y21 = ∂2 (η 12 ∂2 H + (1 + κ)u1 ∂2 K),
Y12 = ∂1 (η 12 ∂1 H + η 22 ∂2 H + 2u2 ∂2 K + (1 + κ)u1 ∂1 K − K),
Y22 = ∂2 (η 12 ∂1 H + η 22 ∂2 H + 2u2 ∂2 K + (1 + κ)u1 ∂1 K − K). Choosing H and K such that η 12 ∂2 H + (1 + κ)u1 ∂2 K = F,
η 12 ∂1 H + η 22 ∂2 H + 2u2 ∂2 K + (1 + κ)u1 ∂1 K − K = G, we get κ
X11 = θ 2 S,
X21 = X12 = X22 = 0.
Finally, taking a suitable choose of H and K, we can also remove S. In particular, we have • for κ 6= 0, −2 κ
(1 + κ)u1 θ1+ 2 S H= , (η 12 )2 κ(κ + 2) • for κ = 0
κ
θ1+ 2 S K = − 12 , η κ(κ + 2)
(2η 12 u2 − η 22 u1 )(log(2η 12 u2 − η 22 u1 ) − 1)u1S , 4(η 12 )2 R Z Z 22 u2 S du1 (2η 12 u2 − η 22 u1 )(log(2η 12 u2 − η 22 u1 ) − 1)S η ∂1 (u1S) 1 1 K= − − du du u1 4η 12 2η 12 u1 H=
• for κ = −2 log(2η 12 u2 + η 22 u1 )u1 S H= , 4(η 12 )2
R S du1 log(2η 12 u2 + η 22 u1 )S K= + . 4η 12 2η 12 u1
Thus, first-order deformations are trivial.
31
A.2 Second order deformations We have seen that in all cases Q1 can be eliminated by a Miura transformation. For this reason, without loss of generality, we can assume the pencil has the form Πλ = ωλ + ǫ2 Q2 + ǫ3 Q3 + ... Using the same arguments applied to first order deformations we can easily prove that • general second order deformations can be always written as Q2 = dω2 X for a suitable vector field of degree 2 X i = X1i (u1 , u2 )u1xx +X2i (u1 , u2 )(u1x )2 +X3i (u1 , u2)u1x u2x +X4i (u1 , u2 )(u2x )2 +X5i (u1 , u2)u2xx , satisfying dω1 dω2 X = 0. • trivial second order deformations are those corresponding to vector fields of the form ω1 δH + ω2 δK, where the hamiltonian functionals H and K have hamiltonian densities of degree 1, namely Z Z 1 2 1 1 2 2 H= h1 (u , u )ux + h2 (u , u )ux dx, K = k1 (u1 , u2 )u1x + k2 (u1, u2 )u2x dx. Before to go into the details of the computations, let us observe that ∂H d ∂H δH ! 1 2 2 δu1 ∂u1 − dx ∂u1x R(u , u )u x = , δH = δH = ∂H d ∂H −R(u1 , u2)u1x − ∂u2 dx ∂u2x δu2
for R(u1 , u2) = ∂1 H2 (u1 , u2) − ∂2 H1 (u1 , u2 ) and similarly ∂K d ∂K δK ! − 1 2 2 δu1 ∂u1 dx ∂u1x S(u , u )u x = , δK = 1 2 1 δK = ∂K d ∂K −S(u , u )ux − ∂u2 dx ∂u2x δu2 for S(u1 , u2) = ∂1 K2 (u1 , u2) − ∂2 K1 (u1 , u2 ). We now proceed as follows:
1. We solve the equation dω1 dω2 X = 0, which leads to a solution depending on two functions of two variables and at most four functions of one variable. 32
2. Up to Miura-type transformations, that is, using the freedom given by the functions R and S, we can eliminate the two functions of two variables. 3. In the cases T3, N3, N5 and N6 with κ 6= −1, −2, we still use a Miura-type transformation to reduce the deformation to a more suitable form (see step 4). 4. The last step is quite straightforward. We firstly take a generic Hamiltonian vector field of the form X = ω1 δH − ω2 δK with Z X Z X i j H= hij ux log ux dx, K= kij uix log ujx dx, i,j
i,j
where the coefficients hij and kij are arbitrary functions of (u1 , u2 ). Then, comparing X with the vector field obtained above (step 3), we get the values of hij and kij which correspond to the final expression written in Theorem 1. Let us discuss in detail each case. In what follows, all the functions Xji , R, S, i = 1, 2, j = 1, . . . , 5, will depend on (u1 , u2), unless stated otherwise. A.2.1 T3. Second order deformations Let us assume η 22 6= 0. The solution of dω1 dω2 X = 0 for deg(X) = 2 is given by X11 = X11 , X21 = X21 , 1 2 X31 = ∂2 X11 − ∂2 X52 , 3 3 1 X4 = 0, X51 = 0, η 22 u1 4 η 22 2 1 1 1 2 2 X1 = X − − ∂ X ∂ (u X ) + 2X 1 1 2 1 5 5 + F1 , η 12 3 3η 12 X22 = ∂1 X12 , X32 = ∂2 X12 + ∂1 X52 , X42 = ∂2 X52 , X52 = F2 e
−η 12 u2 η 22 u1
− X11 .
where F1 , F2 depend on u1 . The components Y i of the vector field Y = ω1 δH+ω2 δK are Y11 = −η 12 R + u1 S,
Y21 = −η 12 ∂1 R + u1 ∂1 S, 33
Y31 = −η 12 ∂2 R + u1 ∂2 S, Y41 = 0,
Y51 = 0, Y12 = −η 22 R,
Y22 = −η 22 ∂1 R,
Y32 = η 12 ∂1 R − η 22 ∂2 R − u1 ∂1 S − S, Y42 = η 12 ∂2 R − u1 ∂2 S,
Y52 = η 12 R − u1 S.
Choosing R and S such that Xi1 = Yi1 for i = 1, 2, we finally obtain X11 = 0 X21 = 0 1 X31 = − ∂2 X52 , 3 1 X4 = 0, X51 = 0, η 22 1 2 2 ∂ (u X ) + 2X 1 5 5 + F1 , 3η 12 = ∂1 X12 ,
X12 = − X22
X32 = ∂2 X12 + ∂1 X52 , X42 = ∂2 X52 , X52 = F2 e
−η 12 u2 η 22 u1
.
Thus, these coefficients depend on two functions F1 , F2 in the variable u1 . In the case η 22 = 0, the computation is easier. The condition dω1 dω2 X = 0 implies X11 = X11 X21 = X21 X31 = ∂2 X11 , X41 = 0, X51 = 0, X12 = F, X22 = ∂1 F, X32 = −∂1 X11 ,
X42 = −∂2 X11 ,
34
X52 = −X11 . where F depends on u1 . Also in this case the freedom in R and S allows us to reduce X11 and X21 to zero, obtaining X 1 = 0,
X 2 = F u1xx + ∂1 F (u1x )2 = (F u1x )x
The second component of the vector field can be written as Z 2 X2 = ∂x F du1, and setting f = F u1 yields Q2 =
! 0 0 . 0 fxx δ ′ + 3fx δ ′′ + 2f δ ′′′
Finally, in order to get the form we need to compute hij (step 3), we perform the canonical Miura transformation generated by the local Hamiltonian Z 22 1 2 ′ u2 η (u ) F2 u2 F2 − ηη12 22 u1 + e H=− u1x dx. 12 2 12 3(η ) 3η S1 Remark. Let us point out that this solution can be obtained from the general case in the limit η 22 → 0. A.2.2 N5. Second order deformations The condition dω1 dω2 X = 0 for deg(X) = 2 implies X11 = X11 , X21 = ∂1 X11 , X31 = ∂2 X11 , X41 = 0, X51 = 0, X12 = X12 , 5η 12 − 2η 22 3/2 2 θ F2 + θ(η 22 X11 − η 12 X12 + F1 ), X22 = ∂1 X12 + θ1/2 ∂1 F2 + 3 3 4η 12 − 3η 22 3/2 X32 = ∂2 X12 − ∂1 X11 + θ1/2 ∂1 F2 − θ F2 , 6 X42 = −η 12 θ3/2 F2 − ∂2 X11 , X52 = θ1/2 F2 − X11 ,
35
where Fi , for i = 1, 2, are functions depending on u1 and θ = (2η 12 (u1 + u2 ) − η 22 u1 )−1 . The components Y i of the vector field Y = ω1 δH + ω2 δK are Y11 = −(η 12 R + u1 S),
Y21 = −∂1 (η 12 R + u1 S),
Y31 = −∂2 (η 12 R + u1 S),
Y41 = 0,
Y51 = 0, Y12 = −(η 22 R + 2(u1 + u2 )S),
Y22 = −(η 22 ∂1 R + 2(u1 + u2 )∂1 S + S),
Y32 = ∂1 (η 12 R + u1 S) − ∂2 (η 22 R + 2(u1 + u2 )S), Y42 = ∂2 (η 12 R + u1 S),
Y52 = η 12 R + u1 S. Choosing R, S such that X1i = Y1i for i = 1, 2, we can reduce X 1 to zero and the coefficients of X 2 respectively to X12 = 0, 7 2 X22 = ∂1 (θ1/2 F2 ) − ∂2 (θ1/2 F2 ) − η 22 θ3/2 F2 + θF1 , 3 3 1 X32 = ∂1 (θ1/2 F2 ) − ∂2 (θ1/2 F2 ), 3 2 1/2 X4 = ∂2 (θ F2 ), X52 = θ1/2 F2 . Thus, the deformations of degree 2 depend on two functions of u1 . To reduce the deformation in the form written in Theorem 1 (step 3) we perform the canonical Miura transformation generated by 22 Z u1 (3η − 8η 12 )θ1/2 F2 log(θ−1 )F1 −1/2 ′ H= u1x dx. +θ F2 + 12 2 (η ) 6 2 S1 A.2.3 N3, N4 and N6. Second order deformations The vector fields Y = P δH + QδK are given by Y11 = −(η 12 R + (1 + κ)u1 S),
Y21 = −∂1 (η 12 R + (1 + κ)u1 S) + κS,
Y31 = −∂2 (η 12 R + (1 + κ)u1 S),
Y41 = 0,
36
Y51 = 0, Y12 = −(η 22 R + 2u2 S),
Y22 = −∂1 (η 22 R + 2u2 S),
Y32 = ∂1 (η 12 R + (1 + κ)u1 S) − ∂2 (η 22 R + 2u2 S), Y42 = ∂2 (η 12 R + (1 + κ)u1 S), Y52 = η 12 R + (1 + κ)u1 S. In studynig the solutions of the equation dω1 dω2 X = 0 we have to distinguish 3 cases: κ = 0, κ = −2, κ 6= 0, 2. This is due to the fact that conditions coming from this equation include the following: κ(κ + 2)X51 (u1 , u2 ) = 0. Case 1: κ = 0. The condition dω1 dω2 X = 0 for deg(X) = 2 leads to X11 = X11 , X21 = ∂1 X11 + θF1 , X31 = θ∂1 F2 − η 22 θ2 F2 + ∂2 X11 ,
X41 = 2η 12 θ2 F2 , X51 = θF2 , X12 = X12 ,
X22 = ∂1 X12 + θF3 , 1 η 22 3 ∂12 F2 2 2 θ 2 F4 − ∂1 X11 + ∂2 X12 , X3 = − 12 + θ ∂1 F4 − η 2 3
X42 = η 12 θ 2 F4 − ∂2 X11 , 1 ∂1 F2 X52 = − 12 + θ 2 F4 − X11 , η where Fi for i = 1, . . . , 4 are arbitrary functions depending on u1 , and θ = (η 22 u1 − 2η 12 u2 )−1 . Choosing R and S such that X1i = Y1i for i = 1, 2, we can reduce both X1i , i = 1, 2, to zero, obtaining X11 = 0, X21 = θF1 , X31 = ∂1 (θF2 ) X41 = ∂2 (θF2 ), X51 = θF2 , X12 = 0, 37
X22 = θF3 , 1 ∂1 F2 2 X3 = ∂1 θ 2 F4 − 12 , η 1 ∂1 F2 2 X4 = ∂2 θ 2 F4 − 12 , η 1 ∂1 F2 X52 = θ 2 F4 − 12 . η In this case, the deformations of degree 2 depend on four functions on u1 . Case 2: κ = −2. The condition dω1 dω2 X = 0 for deg(X) = 2 implies X11 = X11 , 4(η 22 )2 θ4 F2 − 2η 22 θ3 ∂1 F2 η 12 12 2 22 1 +2η θX1 − 2η θX1 + θF1 , 5
X21 = ∂1 X11 + 2η 22 θ 2 F4 +
5
X31 = ∂2 X11 − θ3 ∂1 F2 + 3η 22 θ4 F2 + 2η 12 θ 2 F4 ,
X41 = −4η 12 θ4 F2 , X51 = θ3 F2 ,
X12 = X12 , X22 = ∂X12 + F3 , X32 = ∂2 X12 − ∂1 X11 +
X42 X52
4η 22 θ3 ∂1 F2 − θ2 ∂12 F2 − 6(η 22 )2 θ4 F2 η 12
5 3 3 +θ 2 ∂1 F4 − η 22 θ 2 F4 , 2 5 3 = 4θ ∂1 F2 − 12η 22 θ4 F2 − 3η 12 θ 2 F4 − ∂2 X11 , 3 2η 22 θ3 F2 − θ2 ∂1 F2 − X11 + θ 2 F4 , = 12 η
here θ = (2η 12 u2 + η 22 u1 )−1 and Fi = Fi (u1 ), for i = 1, . . . , 4. Choosing R, S such that X1i = Y1i for i = 1, 2, we can reduce X1i to zero, obtaining X11 = 0, X21
∂1 (θ2 F2 ) = 2η θ θ F4 − η 12 22
3 2
5
X31 = 2η 12 θ 2 F4 − ∂1 (θ3 F2 ),
+ θF1 ,
X41 = −4η 12 θ4 F2 , X51 = θ3 F2 ,
X12 = 0, X22 = F3 , 38
X32
∂12 (θ2 F2 ) = ∂1 (θ F4 ) − , η 12 3 2
3
X42 = 4∂1 (θ3 F2 ) + ∂2 (θ 2 F4 ), 3 ∂1 (θ2 F2 ) . X52 = θ 2 F4 − η 12 Also in this case, the deformations depend on four functions on u1 . Case 3: κ 6= 0, −1, −2. The condition dω1 dω2 X = 0 for deg(X) = 2 implies X11 = X11 , κ(κ + 2) κ−1 κ(κ2 + 7κ + 4)η 22 κ−3 2 ∂ F − θ θ 2 F2 1 2 3(κ + 1)2 6(κ + 1) +θ−1 (κ(η 22 X11 − η 12 X12 ) + F1 ), κ(κ − 1)η 12 κ−3 θ 2 F2 , = ∂2 X11 − 3(κ + 1) = 0,
X21 = ∂1 X11 +
X31 X41
X51 = 0, X12 = X12 , X22 = ∂2 X12 , κ−3 1 ∂1 F2 − η 22 (κ − 1)(κ + 1)θ 2 F2 , 2 1 12 κ−3 2 = (κ − 1)η θ F2 − ∂1 X1 ,
X32 = ∂2 X12 − ∂1 X11 + θ X42
X52 = θ
κ−1 2
κ−1 2
F2 − X11 ,
here θ = 2η 12 u2 − (κ + 1)η 22 u1 and Fi for i = 1, 2 are arbitrary functions depending on u1 . Choosing R, S such that X1i = Y1i for i = 1, 2 we can remove X1i , obtaining X11 = 0, κ(κ + 2) κ−1 κ(κ2 + 7κ + 4)η 22 κ−3 2 ∂ F − θ θ 2 F2 + θ−1 F1 , X21 = 1 2 3(κ + 1)2 6(κ + 1) κ−1 κ X31 = − ∂2 (θ 2 F2 ), 3(κ + 1) 1 X4 = 0, X51 = 0, X12 = 0, X22 = 0, X32 = ∂1 (θ
κ−1 2
X42 = ∂2 (θ
κ−1 2
X52 = θ
κ−1 2
F2 ), F2 ),
F2 . 39
In this last case, the deformations depend on two functions of u1 . The canonical Miura transformation reducing the pencil to the form described in the step 3 is generated by the Hamiltonian functional κ−1 2
(4κη 12 (κ − 1)u2 + η 22 (κ + 1)(2κ3 + 7κ2 + 12κ + 3)u1 )F2 6(η 11 )2 (κ + 1)2 (κ − 1) S1 ! κ+1 θ 2 (2κ + 3)u1 F2′ log θ(κ + 1)u1 F1 u1x dx. + + 11 2 2 11 2 3(η ) (κ + 1) 2(η ) κ
H=
Z
−
θ
B Appendix. Lift of Frobenius structures Recall that a Frobenius manifold is a smooth manifold M equipped with a pseudometric g with Levi-Civita connection ∇, a symmetric bilinear tensorial product on vector fields ·, and two vector fields e, E such that • ∇λX Y = ∇X Y + λX · Y defines a flat affine connection ∇λ for all λ ∈ R, • ∇e = 0, [e, E] = e, and e · X = X for all vector fields X, • ∇(∇E) = 0, LE · = ·, and LE g = kg for some constant k. Theorem 11. Let (M, g, ·, e, E) be a Frobenius manifold. Then the lifted tensors gˆ,ˆ·, eˆ, Eˆ define a structure of Frobenius manifold on T M. The Frobenius potential of the lifted ∂F structure is given by the lift of the Frobenius potential Fˆ = v i ∂u i. Proof: From (4.2) one readily sees that gˆ is symmetric and non-degenerate as soon as g is. ˆ is the Levi-Civita connection If ∇ is the Levi-Civita connection of g, then the lift ∇ of gˆ. This follows by uniqueness of Levi-Civita connection once one noticed that ˆ g = 0 and that ∇ ˆ is torsion free. To see this notice that ∇ˆ ˆ g = 0 for ∇g = 0, and ∇ˆ ˆ is torsion free by Proposition 8 and by torsion-freeness of ∇. that ∇ From (4.4) is clear that ˆ· is symmetric for · is. Moreover, by definition of comˆλ Y = ∇ ˆ X Y + λXˆ·Y for all λ ∈ R, where plete lift for connections it follows that ∇ X ˆλ now X, Y are arbitrary tensor fields on T M. Thanks to Proposition (8), then ∇ is flat. All other conditions follows directly from definition of complete lift, and invariance of Lie derivative under complete lift. At this point recall that a Frobenius manifold is said to be massive if the algebra structure induced by the product · on any tangent space to M is semisimple. More explicitly this means that there is no tangent vector X on M such that X · . . .· X = 0 for some finite product. One may wonder whether semisemplicity assumption is 40
preserved by complete lift or not. In fact it is not, nor is possible to get a massive Frobenius manifold by complete lift of any Frobenius structure on M. The reason is that any vector Y which is tangent to the fibers of T M is an idempotent for the algebra structure induced by ˆ·. Indeed any such vector has the local expression Y i ∂y∂ i , whence it follows that Y ˆ·Y = 0 thanks to (4.4). Remark. Given a Frobenius manifold (M, g, ·, e, E) one can define a hierarchy of quasilinear systems of PDEs of the form uitp,α = P ij
δHp,α , δuj
i = 1, ..., n, p = 1, ..., n, α = 0, 1, 2, 3, ...
where P ij is Hamiltonian operator of hydrodynamic type associated with the invariant metric g and Hp,α are suitable local functionals in involution Z δHp,α ij ij k δHq,β {Hp,α , Hq,β }P = dx = 0 g ∂ + b u x x k i δuj S 1 δu
with respect to the associated Poisson bracket {, }P . It is easy to check that the flows of the lifted hierarchy ˆ p,α δH , uitp,α = Pˆ ij δuj
i = 1, ..., 2n, p = 1, ..., n, α = 0, 1, 2, 3, ...
coincide with "half " of the flows of the principal hierarchy of the lifted Frobenius structure. The involutivity of the lifted Hamiltonian functionals Z ˆ Hp,α = v s ∂s hp,α dx S1
follows from the identity (4.9). Indeed, due to this identity any family of 1-forms in involution with respect to {·, ·}P defines a family of Hamiltonians in involution with respect to {·, ·}Pˆ . If the 1-forms are exact the Hamiltonians on the tangent bundle are the lift of the Hamiltonians on the base manifold.
C Appendix. Lift of Hamiltonian vector fields R Given a Hamiltonian vector field P δH with S 1 h(u, ux , ...) dx, we want to compare its complete lift X ∂(P δH ) ∂ δH ∂ δu [ + v(k) P δH = P δu ∂u ∂u(k) ∂v k
with the vector field
ˆ = P δH ∂ + Pˆ δ H δu ∂u
ˆ X ˆ ∂P δ H δH P + v(t) δu ∂u(t) δv t 41
!
∂ ∂v
ˆ v] = where H[u, show that
R
S1
dx. Since the components along v δH δu P
We observe that
∂ ∂u
coincide we have to
X ˆ X ˆ ) ∂(P δH ∂P δ H δH δu v(k) . + v(t) = δu ∂u δv ∂u (t) (k) t k
ˆ δH δH = , δv δu
ˆ δH δ = δu δu
XZ k
! ∂h v(k) dx , ∂u(k) S1
where the second identity has been obtained integrating by parts. Using these facts P and taking into account that the operators ∂x and k v(k) ∂u∂(k) commute, we get ˆ X ˆ δH ∂P δ H + v(k) = δu ∂u(k) δv k ! Z X ∂h ∂P δH δ X v(k) dx + v(k) P = δu ∂u(k) ∂u(k) δu S1 k k X X ∂P δH ∂2h h h + v(k) = P (−1) ∂x v(k) ∂u(k) ∂u(h) ∂u(k) δu k h,k " # X X X ∂ ∂P δH ∂h P v(k) (−1)h ∂xh + v(k) = ∂u(k) ∂u(k) ∂u(k) δu
P
k
X k
v(k)
h
k
δH ) δu
∂(P . ∂u(k)
In the non scalar case the proof works in exactly the same way.
Ackowledgements We would like to thank Jenya Ferapontov and Raffaele Vitolo for useful discussions and Joseph Krasil’shchik and Alik Verbovetsky for pointing out the reference [23]. P.L. is partially supported by the Italian MIUR Research Project Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite and by GNFM Progetto Giovani 2014 Aspetti geometrici e analitici dei sistemi integrabili.
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