A note on Poisson-Lie algebroids I Liviu Popescu Abstract. In this paper we generalize the linear contravariant connection on Poisson manifolds to Lie algebroids and study its tensors of torsion and curvature. A Poisson connection which depends only on the Poisson bivector and structural functions of the Lie algebroid is given. The notions of complete and horizontal lifts are introduced and their compatibility conditions are pointed out. M.S.C. 2000: 53D17, 17B66, 53C05. Key words: Poisson manifolds, Lie algebroids, contravariant connection, complete and horizontal lifts. 1 Introduction Poisson manifolds were introduced by A. Lichnerowicz in his famous paper [11] and their properties were later investigated by A. Weinstein [19]. The Poisson manifolds are the smooth manifolds equipped with a Poisson bracket on their ring of functions. The Lie algebroid [12] 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 a Lie algebroid [18]. In the last years the various aspects of these subjects have been studied in the different directions of research [18], [15], [3], [4] [13], [10], [1], [9], [2]. In [16] the author has investigated the properties of connections on a Lie algebroid and together with D. Hrimiuc [8] studied the nonlinear connections of its dual. In [3] the linear contravariant Poisson connections on vector bundle are pointed out. The purpose of this paper is to study some aspects of the geometry of the Lie algebroids endowed with a Poisson structure, which generalize some results on Poisson manifolds to Lie algebroids. The paper is organized as follows. In the section 2 we recall the Cartan calculus and the Schouten-Nijenhuis bracket to the level of a 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 the Balkan Journal of Geometry and Its Applications, Vol.14, No.2, 2009, pp. 79-89. c Balkan Society of Geometers, Geometry Balkan Press 2009.
80 Liviu Popescu results of R. Fernandes from [3]. The section 3 deals with the prolongation of Lie algebroid [7] over the vector bundle projection. We study the properties of the complete lift of a Poisson bivector and introduce the notion of horizontal lift. The compatibility conditions of these bivectors are investigated. We remark that in the particular case of the Lie algebroid E = T M, σ = Id some results of G. Mitric and I. Vaisman [15] are obtained. 2 Lie algebroids Let us consider a differentiable, n-dimensional manifold M 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 [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 one 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 given by σs α = σα i x i, [s α, s β ] = L αβ s, i = 1, n, α, β, = 1, m, are called the structure functions of the Lie algebroid and satisfy the structure equations on Lie algebroid 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,
A note on Poisson-Lie algebroids I 81 Locally, if f C M then d E f = f x i σ i αs α, where {s α } is the dual basis of {s α } and if θ ΓE, θ = θ α s α then Particularly, we get d E θ = σα i θ β x i 1 2 θ L αβ sα s β, d E x i = σ i αs α, d E s α = 1 2 Lα βs β s. The Schouten-Nijenhuis bracket on E is given by [18] [X 1... X p, Y 1... Y q ] = 1 p+1 p where X i, Y j ΓE. i=1 j=1 2.1 Poisson structures on Lie algebroids q 1 i+j [X i, Y j ] X 1... ˆ X i... X p Y 1... Y ˆ j... Y q Let us consider the bivector i.e. contravariant, skew-symmetric, 2-section Π Γ 2 E given by the expression 2.1 Π = 1 2 παβ xs α s β. Definition 2.1 The bivector Π is a Poisson bivector on E if and only if the relation [Π, Π] = 0, is fulfilled. Proposition 2.1 Locally, the condition [Π, Π] = 0 is expressed as 2.2 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 a Lie algebroid with Poisson structure. A corresponding Poisson bracket on M is given by {f 1, f 2 } = Πd E f 1, d E f 2, f 1, f 2 C M. 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,
82 Liviu Popescu 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, and it results that E, [.,.] π, σ is a Lie algebroid [13]. 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. Accordingly, we get the cohomology of Lie algebroid E with the anchor σ and the bracket [.,.] π which generalize the Poisson cohomology of Lichnerowicz for Poisson manifolds [11]. In the following we deal with the notion of contravariant connection on Lie algebroids, which generalize the similar notion on Poisson manifolds [18], [3]. Definition 2.2 If ρ, θ ΓE and Φ, Ψ ΓE then the linear contravariant connection on a Lie algebroid 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 2.3 The torsion and curvature of the linear contravariant connection are given by T ρ, θ = D ρ θ D θ ρ [ρ, θ] π, where ρ, θ, µ ΓE. Rρ, θµ = D ρ D θ µ D θ D ρ µ D [ρ,θ]π θ, In the local coordinates we define the Christoffel symbols Γ αβ considering D s αs β = Γ αβ s, and under a change of coordinates x i = x i x i, i, i = 1, n on M, and y α = A α α y α, α, α = 1, m on E, corresponding to a new base s α = A α α s α, these symbols transform according to
A note on Poisson-Lie algebroids I 83 2.3 Γ α β = Aα α A β β A Γαβ + A α α A A β σi ε x i παε. Proposition 2.2 The local components of the torsion and curvature of the linear contravariant connection on a Lie algebroid have the expressions T αβ ε = Γ αβ ε Γ βα ε π α L β ε + π β L α ε σε x i, R αβ δ = Γ αε δ Γ β ε Γ βε δ Γα ε +π αε σε 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 the following relations are fulfilled 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. 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 the contravariant derivative is given by D θ T = θ α T i1...ir j 1...j s / α s i1 s ir s j 1 s j s, where T i1...ir j 1...j s / α = π αε σε i T i 1...i r j 1...j s x i + r Γ i a s α ε T i1...ε...ir j 1...j s Γ εα jb T i1...ir j 1...ε...j s, a=1 b=1 and / denote the contravariat derivative operator. We recall that a tensor field T on E is called parallel if and only if DT = 0. Definition 2.4 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 Cristoffel symbols Γ αβ associated contravariant derivative. We obtain: and Proposition 2.3 The contravariant connection D with the coefficients given by 2.4 Γ αβ is a Poisson connection. = Γ αβ 1 2 π επ αε / β, Proof. Considering / the contravariant derivative operator with respect to the contravariant connection D, we get π β / α = π αε σε i π β x i + Γ βα ε π ε + Γ α ε π βε =
84 Liviu Popescu = π αε σε i π β x i + Γ βα ε 1 2 π ετ π βτ / α π ε + Γ α ε 1 2 π ετ π τ / α π βε = 0, which concludes the proof. Remark 2.1 Considering the expression 2.5 Γ αβ = σ x i, in 2.4 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 the Lie algebroid. Proof. Under a change of coordinates, the structure functions σα i change by the rule [17], [14] σα i Aα α = xi x i σi α, and, by direct computation, follows that the coefficients 2.5 satisfy the transformation law 2.3. Theorem 2.1 If the following relation π αβ π δ L ε β = 0, α,ε,δ is true, then the connection D with the coefficients Γ αβ is a Poisson connection on Lie algebroid. = σ x i, Proof. Using relation 2.2 we obtain that π β / α = 0 if and only if the required relation is fulfilled. Proposition 2.4 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
A note on Poisson-Lie algebroids I 85 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 π β / α + 1 2 πθ X βα θ 1 2 πβθ X α 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 = + 1 2 πβθ X α θ 1 2 πθ X βα θ = 0, because π β / α = 0, which ends the proof. 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 [7], [14], [10], [16]. 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. For s ΓE we can consider the vertical lift of s given by s v u = ϕsπu, for u E, where ϕ : 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 [5] [6]. 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 [14] s v u = 0, s v u, s c u = sπu, s c u, u E. Another canonical object on T E is the Euler section C, which is the section of T E E defined by Cu = 0, ϕu 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
86 Liviu Popescu 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 a section ρ is α and therefore In particular ρ c = ρ α X α + ρ α L α βρ β y V α, σ 1 ρ c = ρ α σα i x i + ρ α σi x i Lα βρ β 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 X α + i L y V α, where{x α, V α } denotes the corresponding dual basis of {X α 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 βx β X, d E V α = 0. A nonlinear connection N on T E [17] 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 The sections h = 1 id + N, 2 v = 1 id N. 2 δ α = X α h = X α N β α V β, generate a basis of HT E, where Nα β 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 Nβ αx β. The Lie brackets of the adapted basis {δ α, V α } are [16] 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,
A note on Poisson-Lie algebroids I 87 3.1 R αβ = δ βn α δ α N β + Lε αβn 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 3.1 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 2.1. We obtain: Proposition 3.1 The complete lift of Π on T E is given by 1 3.2 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 1 X α V β + 2 σi παβ x π δβ L α i δ y V α V β. Proposition 3.2 The complete lift Π c is a Poisson bivector on T E. Proof. Using the relations 3.2 and 2.2, by straightforward computation, we obtain that [Π c, Π c ] = 0, which ends the proof. Proposition 3.3 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. Proof. A direct computation in local coordinates yields L C Π c = L y ε V ε π αβ X α V β + 1 2 σi παβ x i π δβ L α δ y V α V β = L y ε V ε π αβ X α V β π αβ X α V β + L y ε V ε 1 2 σi παβ x y V i α V β y V α V β L y ε V ε π δβ L α δ y V α V β + π δβ L α δ y V α V β 1 2 σi παβ x i = π αβ X α V β 1 2 σi παβ x y V i α V β + π δβ L α δ y V α V β = Π c
88 Liviu Popescu Definition 3.1 Let us consider a Poisson bivector on E given by 2.1 then the horizontal lift of Π to T E is the bivector defined by Π H = 1 2 παβ xδ α δ β. Proposition 3.4 The horizontal lift Π H is a Poisson bivector if and only if Π is a Poisson bivector on E and the following relation is fulfilled. π αβ π δ R ε β = 0, Proof. The Poisson condition [Π, Π] = 0 leads to the relation 2.2 and equation [Π H, Π H ] = 0 yields ε,δ,α π αβ π δ L ε β + π αβ σ i β π εδ x i δ ε δ α δ δ + π αβ π δ R ε βv ε δ α δ = 0, which ends the proof. We recall that two Poisson structures are called 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 3.5 The Poisson bivector Π H Π c if and only if the following relations hold is compatible with the complete lift where we have denoted π rβ π αs N r y s π rs δ r a αβ a lα N β r y l N s y r πr π sα L β sr = 0, + a lβ N α r y l π θβ R α rθ + π εβ L θ ε π εθ L β εy N α r 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 a αβ = σε x i yε + Nε α π εβ Nε β π εα. = 0. Acknowledgements. The author wishes to express his thanks to the organizers of the International Conference DGDS-2008 and MENP-5, August 28 - September 2, 2008, where this paper has presented. Also, many thanks to the referee for useful remarks concerning this paper.
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