POISSON STRUCTURES ON LIE ALGEBROIDS

POISSON STRUCTURES ON LIE ALGEBROIDS LIVIU POPESCU University of Craiova, Dept. of Applied Mathematics in Economy 13 A.I.Cuza st., 200585, Craiova, Romania e-mail: liviupopescu@central.ucv.ro, liviunew@yahoo.com Abstract. In this paper the properties of Lie algebroids with Poisson structures are investigated. We generalize some results of Fernandes [1] regarding linear contravariant connections on Poisson manifolds at the level of Lie algebroids. In the last part, the notions of complete and horizontal lifts on the prolongation of Lie algebroid are studied and their compatibility conditions are pointed out. Key words: Poisson manifolds, Lie algebroids, contravariant connection, complete and horizontal lifts. Mathematics Subject Classification 2000: 53D17, 17B66, 53C05 1. Introduction Poisson manifolds were introduced by A. Lichnerowicz in his famous paper [8] and their properties were later investigated by A. Weinstein [15]. The Poisson manifolds are the smooth manifolds equipped with a Poisson bracket on their ring of functions. In the last years a lot of papers deal with the study of various aspects of this subject in the different directions of research [14], [12], [1]. The Lie algebroid [9] is a generalization of a Lie algebra and integrable distribution. In fact, a Lie algebroid is a vector bundle with a Lie bracket on his space of sections whose properties are very similar to those of a tangent bundle. We remark that the cotangent bundle of a Poisson manifold has a natural structure of Lie algebroid. The purpose of this paper is to study some aspects of Lie algebroids geometry endowed with a Poisson structures, which generalize the Poisson manifolds. The paper is organized as follows. In the section 2 we recall the Cartan calculus and the Schouten-Nijenhuis bracket at the level of on Lie algebroids and introduce the Poisson structure on Lie algebroid. We investigate the properties of linear contravariant connection and its tensors of torsion and curvature. In the last part of this section we find a Poisson connection which depends only on the Poisson bivector and structural functions of Lie algebroid which generalize some results of Fernandes from [1]. The section 3 deals with the prolongation of Lie algebroid [5] over the vector Date: July 30, 2008. 1

2 LIVIU POPESCU bundle projection. We study the properties of the complete lift of a Poisson bivector and introduce the notion of horizontal lift. Finally, the compatibility conditions of these bivectors are investigated. We remark that in the particular case of standard Lie algebroid E = T M some results of Mitric and Vaisman [12] are obtained. 2. Lie algebroids Let M be a differentiable, n-dimensional manifold and T M, π M, M its tangent bundle. A Lie algebroid over the manifold M is the triple E, [, ], σ where π : E M is a vector bundle of rank m over M, whose C M- module of sections ΓE is equipped with a Lie algebra structure [, ] and σ : E T M is a bundle map called the anchor which induces a Lie algebra homomorphism also denoted σ from ΓE to χm, satisfying the Leibnitz rule 1 [s 1, fs 2 ] = f[s 1, s 2 ] + σs 1 fs 2 for every f C M and s 1, s 2 ΓE. Therefore, we get [σs 1, σs 2 ] = σ[s 1, s 2 ], [s 1, [s 2, s 3 ]] + [s 2, [s 3, s 1 ]] + [s 3, [s 1, s 2 ]] = 0. If ω k E then the exterior derivative d E ω k+1 E is given by the formula d E ωs 1,..., s k+1 = k+1 1 i+1 σs i ωs 1,..., ŝi,..., s k+1 + i=1 + 1 i<j k+1 1 i+j ω[s i, s j ], s 1,..., ŝi,..., ˆ s j,...s k+1. where s i ΓE, i = 1, k + 1, and it results that d E 2 = 0. Also, for ξ ΓE on can define the Lie derivative with respect to ξ by L ξ = i ξ d E + d E i ξ, where i ξ is the contraction with ξ. If we take the local coordinates x i on an open U M, a local basis {s α } of sections of the bundle π 1 U U generates local coordinates x i, y α on E. The local functions σαx, i L αβ x on M are called the structure functions of the Lie algebroid. They are given by σs α = σα i x i, [s α, s β ] = L αβ s, i = 1, n, α, β, = 1, m, and satisfy the so called structure equations on Lie algebroid 2 σα j σβ i x j σ i σj α β x j = σi L αβ, σα i L δ β x i + L δ αηl η β = 0. α,β,

POISSON STRUCTURES ON LIE ALGEBROIDS 3 Locally, if f C M then d E f = f σ x αs i α and if θ ΓE, θ = θ i α s α then d E θ = σα i θ β x i 1 2 θ L αβ sα s β, where {s α } is the dual basis of {s α }. Particularly d E x i = σ i αs α, d E s α = 1 2 Lα β sβ s. The Schouten-Nijenhuis bracket is given by [14] p q [X 1... X p, Y 1... Y q ] = 1 p+1 1 i+j [X i, Y j ] X 1... i=1 j=1 ˆ X i... X p Y 1... Y ˆ j... Y q 2.1. Poisson structures on Lie algebroids. Let us consider the bivector i.e. contravariant, skew-symmetric, 2-section Π Γ 2 E given by 3 Π = 1 2 παβ xs α s β. Definition 1. The bivector Π is a Poisson bivector on E if and only if [Π, Π] = 0, where [, ] is Schouten-Nijenhuis bracket. Proposition 1. Locally the condition [Π, Π] = 0 is expressed as 4 α,ε,δ π αβ σβ i π εδ x i + π αβ π δ L ε β = 0 If Π is a Poisson bivector then the pair E, Π is called the Lie algebroid with Poisson structure. The Poisson bracket on E is given by {f 1, f 2 } = Πd E f 1, d E f 2, f 1, f 2 C E We also have the bundle map π # : E E defined by Let us consider the bracket π # ρ = i ρ Π, ρ ΓE. [ρ, θ] π = L π # ρθ L π # θρ d E Πρ, θ, where L is Lie derivative and ρ, θ ΓE. With respect to this bracket and the usual Lie bracket on vector fields, the map σ : E T M given by is a Lie algebra homomorphism σ = σ π #, σ[ρ, θ] π = [ σρ, σθ]. The bracket [.,.] π satisfies also the Leibnitz rule [ρ, fθ] π = f[ρ, θ] π + σρfθ,

4 LIVIU POPESCU and it results that E, [.,.] π, σ is a Lie algebroid [6]. Next, we can define the contravariant exterior differential d π : k E k+1 E by d π ωs 1,..., s k+1 = k+1 1 i+1 σs i ωs 1,..., ŝi,..., s k+1 + i=1 + 1 i<j k+1 1 i+j ω[s i, s j ] π, s 1,..., ŝi,..., ˆ s j,...s k+1. In fact, is obtained the cohomology of Lie algebroid E with the anchor σ and the bracket [.,.] π which generalize the Poisson cohomology of Lichnerowicz for Poisson manifolds [8]. Definition 2. If ρ, θ ΓE and Φ, Ψ ΓE then the linear contravariant connection is an application D : ΓE ΓE ΓE which satisfies the relations i D ρ+θ Φ = D ρ Φ + D θ Φ, ii D ρ Φ + Ψ = D ρ Φ + D ρ Ψ, iii D fρ Φ = fd ρ Φ, iv D ρ fφ = fd ρ Φ + σρfφ, f C M. Definition 3. The torsion and curvature of linear contravariant connection are given by T ρ, θ = D ρ θ D θ ρ [ρ, θ] π, Rρ, θµ = D ρ D θ µ D θ D ρ µ D [ρ,θ]π θ, where ρ, θ, µ ΓE. In the local coordinates we define the Christoffel symbols Γ αβ and we obtain that D s αs β = Γ αβ s, Proposition 2. The local components of torsion and curvature of linear contravariant connection are R αβ δ = Γ αε δ T αβ ε = Γ αβ ε Γβ ε Γ βε δ Γα ε Γ βα ε +π αε σ i ε π α L β ε + π β L α ε σε x i, Γ β δ x i πβε σε i Γ α δ x i +πβν L α νε π αν L β νε σε x i Γε δ. The contravariant connection induces a contravariant derivative D α : ΓE ΓE such that D f1 α 1 +f 2 α 2 = f 1 D α1 + f 2 D α2, f i C M, α i ΓE, D ρ fφ = fd ρ Φ + σρfφ, f C M, ρ, θ ΓE. by

POISSON STRUCTURES ON LIE ALGEBROIDS 5 Let T be a tensor of type r, s with the components T i 1...i r j 1...j s and θ = θ α s α a section of E. The local coordinates expression of contravariant derivative is given by where D θ T = θ α T i 1...i r j 1...j s / α s i1 s ir s j 1 s j s, T i 1...i r j 1...j s / α = π αε σε i T i 1...i r j 1...j s x i + r a=1 Γ i aα ε T i 1...ε...i r j 1...j s and / denote the contravariat derivative operator. A tensor field T on E is called parallel if and only if DT = 0. s b=1 Γ εα j b T i 1...i r j 1...ε...j s, Definition 4. A contravariant connection D is called a Poisson connection if the Poisson bivector is parallel with respect to D. Let us consider a contravariant connection D with the coefficients Γ αβ. We have Proposition 3. The contravariant connection D with the coefficients given by 5 Γ αβ is a Poisson connection. = Γ αβ 1 2 π επ αε / β, Proof. Considering / the contravariant derivative operator with respect to contravariant connection D, we get π β / α = π αε σε i π β x i = π αε σε i π β x i + Γ βα ε 1 2 π ετ π βτ / α π ε + + Γ βα ε π ε + Γ α ε π βε = Γ α ε 1 2 π ετ π τ / α π βε = π β / α 1 2 πβ / α 1 2 πβ / α = 0. Remark 1. Considering Γ αβ = σ x i in relation 5 we obtain a Poisson connection D with the coefficients Γ αβ = σ x i 1 2 π επ αε / β which depends only on the Poisson bivector and structural functions of Lie algebroid.

6 LIVIU POPESCU Theorem 1. The connection D with the coefficients Γ αβ = σ x i, is a Poisson connection if and only if π αβ π δ L ε β = 0. α,ε,δ Proof. Using relation 4 we obtain that π β / α = 0 if and only if the required relation is fulfilled. Proposition 4. The set of Poisson connections on Lie algebroid is given by Γ αβ = Γ αβ + Ω αε νx νβ ε, where Ω αε ν = 1 δ α 2 ν δ ε π ν π αε, and DΓ αβ is a Poisson connection with Xε δβ an arbitrary tensor. Proof. By straightforward computation it results π β / α = π αε σε i π β x i π β / α + 1 2 πε δ β ν δ θ ε π εν π βθ X να θ + Γ βα ε π ε + Γ α ε π βε = + 1 2 πβε δ ν δ θ ε π εν π θ X να θ = π β / α + 1 2 πθ X βα θ 1 2 πβθ X α θ because π β / α = 0, which ends the proof. + 1 2 πβθ X α θ 1 2 πθ X βα θ = 0, 3. The prolongation of Lie algebroid over the vector bundle projection Let E, π, M be a vector bundle. For the projection π : E M we can construct the prolongation of E see [5], [10], [7], [13]. The associated vector bundle is T E, π 2, E where T E = w E T w E with T w E = {u x, v w E x T w E σu x = T w πv w, πw = x M}, and the projection π 2 u x, v w = π E v w = w, where π E : T E E is the tangent projection. The canonical projection π 1 : T E E is given by π 1 u, v = u. The projection onto the second factor σ 1 : T E T E, σ 1 u, v = v will be the anchor of a new Lie algebroid over manifold E. An element of T E is said to be vertical if it is in the kernel of the projection π 1. We will denote V T E, π 2 V T E, E the vertical bundle of T E, π 2, E. If f C M we will denote by f c and f v the complete and vertical lift to E of f defined by f c u = σuf, f v u = fπu, u E.

POISSON STRUCTURES ON LIE ALGEBROIDS 7 For s ΓE we can consider the vertical lift of s given by s v u = sπu v u, for u E, where v u : E πu T u E πu is the canonical isomorphism. There exists a unique vector field s c on E, the complete lift of s satisfying the two following conditions: i s c is π-projectable on σs, ii s c α = L s α, for all α ΓE, where αu = απuu, u E see [3] [4]. Considering the prolongation T E of E over the projection π, we may introduce the vertical lift s v and the complete lift s c of a section s ΓE as the sections of T E E given by see [10] s v u = 0, s v u, s c u = sπu, s c u, u E. Other canonical object on T E is the Euler section C, which is the section of T E E defined by Cu = 0, u v u for all u E. The local basis of ΓT E is given by {X α, V α }, where X α u = s α πu, σα i x i, V α u = 0, u y α, u and / x i, / y α is the local basis on T E. The structure functions of T E are given by the following formulas σ 1 X α = σα i x i, σ1 V α = y α, [X α, X β ] = L αβ X, [X α, V β ] = 0, [V α, V β ] = 0. The vertical lift of a section ρ = ρ α s α and the corresponding vector field are ρ v = ρ α V α and σ 1 ρ v = ρ α y. The expression of the complete lift of α a section ρ is and therefore In particular ρ c = ρ α X α + ρ α L α β ρβ y V α, σ 1 ρ c = ρ α σα i x i + ρ α σi x i Lα β ρβ y y α. s v α = V α, s c α = X α L β αy V β. The coordinate expressions of C and σ 1 C are C = y α V α, σ 1 C = y α y α The local expression of the differential of a function L on T E is d E L = σα i L X α + L x i y V α, where{x α, V α } denotes the corresponding dual basis of α {X α, V α } and therefore, we have d E x i = σαx i α and d E y α = V α. The differential of sections of T E is determined by d E X α = 1 2 Lα β X β X, d E V α = 0.

8 LIVIU POPESCU A nonlinear connection N on T E [11] is an m dimensional distribution called horizontal distribution N : u E HT u E T E that is supplementary to the vertical distribution. This means that we have the following decomposition T u E = HT u E V T u E, for u E. A connection N on T E induces two projectors h, v : T E T E such that hρ = ρ h and vρ = ρ v for every ρ ΓT E. We have h = 1 id + N, 2 v = 1 id N. 2 The sections δ α = X α h = X α N β α V β, generate a basis of HT E, where N β α are the coefficients of nonlinear connection. The frame {δ α, V α } is a local basis of T E called adapted. The dual adapted basis is {X α, δv α } where δv α = V α N α β X β. The Lie brackets of the adapted basis {δ α, V α } are [13] where [δ α, δ β ] = L αβ δ + R αβ V, [δ α, V β ] = N α y β V, [V α, V β ] = 0, 6 R αβ = δ βn α δ α N β + Lε αβ N ε. The curvature of a connection N on T E is given by Ω = N h where h is horizontal projector and N h is the Nijenhuis tensor of h. In the local coordinates we have Ω = 1 2 R αβ X α X β V where R αβ are given by 6 and represent the local coordinate functions of the curvature tensor Ω in the frame 2 T E T E induced by {X α, V α }. 3.1. Compatible Poisson structures. Let us consider the Poisson bivector on Lie algebroid given by relation 3. We obtain Proposition 5. The complete lift of Π on T E is given by 1 7 Π c = π αβ π αβ X α V β + 2 σi x i π δβ L α δ y V α V β. Proof. Using the properties of vertical and complete lifts we obtain Π c = 1 2 παβ s α s β c = 1 2 παβ c s α s β v + 1 2 παβ v s α s β c = = 1 2 π αβ s v α s v β + 1 2 παβ s c α s v β + sv α s c β = 1 2π αβ V α V β + + 1 2 παβ X α L δ αy V δ V β + V α X β L δ β y V δ = = π αβ X α V β + y V α V β. 1 2 σi παβ x i π δβ L α δ Proposition 6. The complete lift Π c is a Poisson bivector on T E.

POISSON STRUCTURES ON LIE ALGEBROIDS 9 Proof. Using the relation 7 by straightforward computation we obtain [Π c, Π c ] = 0 which ends the proof. Proposition 7. The Poisson structure Π c has the following property Π c = L C Π c, which means that T E, Π c is a homogeneous Poisson manifold. Definition 5. Let us consider a Poisson bivector on E given by 3 then the horizontal lift of Π to T E is the bivector defined by Π H = 1 2 παβ xδ α δ β. Proposition 8. The horizontal lift Π H is a Poisson bivector if and only if Π is a Poisson bivector on E and ε,δ,α π αβ π δ R ε β = 0. Proof. The Poisson condition [Π, Π] = 0 leads to the relation 4 and [Π H, Π H ] = 0 yields π αβ π δ L ε β + παβ σβ i π εδ x i δ ε δ α δ δ + π αβ π δ R ε β V ε δ α δ = 0, which ends the proof. Recall that two Poisson structures are compatible if the bivectors ω 1 and ω 2 satisfy the condition [ω 1, ω 2 ] = 0 By straightforward computation in local coordinates we get: Proposition 9. The Poisson bivector Π H is compatible with complete lift Π c if and only if the following relations hold π rβ π αs N r y s π rs N s y r πr π sα L β sr = 0, δ r a αβ a lα N β r y l + a lβ N α r y l π θβ Rrθ α + πεβ L θ ε π εθ L β εy N r α y θ + + σr i π εβ x i y L α ε + π εβ L α εnr π εβ σr i L α ε x i y = 0. where we have denoted a αβ = σε x i yε + Nε α π εβ Nε β π εα.

10 LIVIU POPESCU Acknowledgments I gratefully acknowledge the hospitality and support of the Centre Interfacultaire Bernoulli, Ecole Polytechnique Federale de Lausanne and organizers of the Conference Poisson 2008, where this paper has been presented. Also, I would like to thank Izu Vaisman for helpful comments made about this paper. References [1] R. L. Fernandes, Connections in Poisson geometry I: Holonomy and invariants, J. Differential Geometry, 54 2000 303-365. [2] R. L. Fernandes, Lie Algebroids, Holonomy and Characteristic Classes, Advances in Mathematics, 170 2002 119-179. [3] J. Grabowski, P. Urbanski, Tangent and cotangent lift and graded Lie algebra associated with Lie algebroids, Ann. Global Anal. Geom. 15, 1997, 447-486. [4] J. Grabowski, P. Urbanski, Lie algebroids and Poisson-Nijenhuis structures, Rep. Math. Phys., 40 1997, 195-208. [5] P. J. Higgins, K. Mackenzie, Algebraic constructions in the category of Lie algebroids, Journal of Algebra 129 1990 194-230. [6] Y. Kosmann-Schwarzbach, Poisson manifolds, Lie algebroids, modular classes: a survey, Symmetry, Integrability and Geometry, Methods and Applications, SIGMA 4 2008, 005. [7] M. de Leon, J. C. Marrero, E. Martinez, Lagrangian submanifolds and dynamics on Lie algebroids J. Phys. A: Math. Gen. 38 2005 241-308. [8] A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry, 12 1977 253-300. [9] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, Cambridge, no.124, 1987. [10] E. Martinez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67, 2001 295-320. [11] T. Mestdag, Generalized connections on affine Lie algebroids, Rep. Math. Phys., 51 2003 297-305. [12] G. Mitric, I. Vaisman, Poisson structures on tangent bundles, Diff. Geom and Appl., 18 2003, 207-228. [13] L. Popescu, Geometrical structures on Lie algebroids, Publ. Math. Debrecen 72, 1-2 2008, 95-109. [14] I. Vaisman, Lectures on the geometry of Poisson manifolds, Progress in Math., vol. 118, Birkhäuser, Berlin, 1994. [15] A. Weinstein, The local structure of Poisson manifolds, J. Differential Geometry, 18 1983 523-557