Cohomology Associated to a Poisson Structure on Weil Bundles

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1 International Mathematical Forum, Vol. 9, 2014, no. 7, HIKARI Ltd, Cohomology Associated to a Poisson Structure on Weil Bundles Vann Borhen Nkou 1,2 and Basile Guy Richard Bossoto 1 1 Marien NGOUABI University, Faculty of Science Department of Mathematics BP: 69 - Brazzaville, Congo 2 Abomey Calavi University, IMSP BP: 13, Porto-novo, Benin Copyright c 2014 Vann Borhen Nkou and Basile Guy Richard Bossoto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Let M be a paracompact smooth manifold of dimension n, A a Weil algebra and M A the Weil bundle associated. We define and describe the notion of d-poisson cohomology and of d A -Poisson cohomology on M A. Mathematics Subject Classification: 58A20, 58A32, 17D63, 53D17, 53D05 Keywords: Weil bundle, Weil algebra, Poisson manifold, cohomology 1 Introduction A local algebra in the sens of André Weil or simply a Weil algebra is a real, unitary, commutative algebra of finite dimension with a unique maximal ideal of codimension 1 on R [10]. Let A be a Weil algebra and m be its maximal ideal. We have A = R m.

2 306 Vann Borhen Nkou and Basile Guy Richard Bossoto The first projection A = R m R is a homomorphism of algebra which is surjective, called augmentation and the unique none-zero integer k N such that m k (0) and m k+1 = (0) is the height of A. If M is a smooth manifold, C (M) the algebra of differentiable functions on M and A a Weil algebra of maximal ideal m, an infinitely near point to x M of kind A is a homomorphism of algebras ξ : C (M) A such that [ξ(f) f(x)] m for any f C (M). We denote M A x the set of all infinitely near points to x M of kind A and M A = Mx A. x M The set M A is a smooth manifold of dimension dim M dim A called manifold of infinitely near points of kind A[6]. When both M and N are smooth manifolds and when h : M N is a differentiable application, then the application h A : M A N A,ξ h A (ξ), such that, for any g C (N), [ h A (ξ) ] (g) =ξ(g h) is also differentiable. When h is a diffeomorphism, it is the same for h A. Moreover, if ϕ : A B is a homomorphism of Weil algebras, for any smooth manifold M, the application ϕ M : M A M B,ξ ϕ ξ is differentiable. In particular, the augmentation A R

3 Cohomology associated to a Poisson structure 307 defines for any smooth manifold M, the projection π M : M A M, which assigns every infinitely near point to x M to its origin x. Thus (M A,π M,M) defines the bundle of infinitely near points or simply weil bundle [7],[4],[10]. If (U, ϕ) is a local chart of M with coordinate functions (x 1,x 2,..., x n ), the application U A A n,ξ (ξ(x 1 ),ξ(x 2 ),..., ξ(x n )), is a bijection from U A into an open of A n. The manifold M A is a smooth manifold modeled over A n, that is to say an A-manifold of dimension n [1],[9]. The set, C (M A,A) of differentiable functions on M A with values in A is a commutative, unitary algebra over A.When one identitifies R A with A, for f C (M), the application f A : M A A, ξ ξ(f) is differentiable. Moreover the application C (M) C (M A,A),f f A, is an injective homomorphism of algebras and we have: (f + g) A = f A + g A ;(λ f) A = λ f A ;(f g) A = f A g A for λ R, f and g belonging to C (M). We denote X(M A ), the set of all vector fields on M A. According to [1], We have the following equivalent assertions: 1. X : C (M A ) C (M A ) is a vector field on M A ; 2. X : C (M) C (M A,A) is a linear application which verifies X(fg)=X(f) g A + f A X(g) for any f,g C (M) i.e is a derivation of C (M) intoc (M A,A) with respect to the module structure C (M A,A) C (M) C (M A,A), (ϕ, f) ϕ f A. Thus, the set X(M A ) of all vector fields on M A is a C (M A,A)-module.

4 308 Vann Borhen Nkou and Basile Guy Richard Bossoto When is a vector field on M, the application θ : C (M) C (M) θ A : C (M) C (M A,A),f [θ(f)] A, is a vector field on M A. The vector field θ A is the prolongation to M A of the vector field θ. Theorem 1 If X is a vector field on M A considered as a derivation of C (M) into C (M A,A), then there exists, an unique derivation such that such that 1. X is A-linear; 2. X [ C (M A ) ] C (M A ); X : C (M A,A) C (M A,A) 3. X(f A )=X(f) for any f C (M). Thus, the application [, ]:X(M A ) X(M A ) X(M A ), (X, Y ) X Y Ỹ X, is A-bilinear and defines a structure of Lie algebra over A on X(M A )[1]. The goal of this paper is to define and describe the notion of d-poisson cohomology and of d A -Poisson cohomology. 2 Poisson structure on Weil bundles In this section, M is a Poisson manifold i.e there exists a bracket {, } on C (M) such that the pair (C (M), {, }) is a real Lie algebra and for any f C (M), the application ad(f) :C (M) C (M),g {f,g} is a derivation of commutative algebra i.e {f,g h} = {f,g} h + g {f,h}

5 Cohomology associated to a Poisson structure 309 for f,g,h C (M) [5],[8]. We denote C (M) Der R [C (M)],f ad(f), the adjoint representation and d the operator of cohomology associated to this representation. For any p N, Λ p P ois (M) =Cp [C (M),C (M)] denotes the C (M)-module of skew-symmetric multilinear forms of degree p from C (M) intoc (M). We have Λ 0 P ois (M) =C (M). The A-algebra C (M A,A) is a Poisson algebra over A if there exists a bracket {, } on C (M A,A) such that the pair (C (M A,A), {, }) is a Lie algebra over A satisfying {ϕ 1 ϕ 2,ϕ 3 } = {ϕ 1,ϕ 3 } ϕ 2 + ϕ 1 {ϕ 2,ϕ 3 } for any ϕ 1,ϕ 2,ϕ 3 C (M A,A) [3],[2]. When M is a Poisson manifold with bracket {, }, for any f,g C (M), For any f C (M), let ad(fg)=ad(f) g + f ad(g). [ad(f)] A : C (M) C (M A,A),g {f,g} A, be the prolongation of the vector field ad(f) and let [ad(f)] A : C (M A,A) C (M A,A) be the unique A-linear derivation such that for any g C (M). [ad(f)] A (g A )=[ad(f)] A (g) ={f,g} A Theorem 2 [3] For ϕ C (M A,A), the application is a vector field on M A. τ ϕ : C (M) C (M A,A),f [ad(f)] A (ϕ)

6 310 Vann Borhen Nkou and Basile Guy Richard Bossoto We denote τ ϕ : C (M A,A) C (M A,A) the unique A-linear derivation such that τ ϕ (f A )=τ ϕ (f) for any f C (M). We have for f C (M), τ f A = and for ϕ, ψ C (M A,A) and for a A, For any ϕ, ψ C (M A,A), we let [ad(f)] A, τ ϕ+ψ = τ ϕ + τ ψ ; τ a ϕ = a τ ϕ ; τ ϕ ψ = ϕ τ ψ + ψ τ ϕ. {ϕ, ψ} A = τ ϕ (ψ). In [3] we show that this bracket defines a structure of A-Poisson algebra on C (M A,A). Theorem 3 If M is a Poisson manifold with bracket {, }, then {, } A is the prolongation on M A of the structure of Poisson on M defined by {, }. 3 d-poisson cohomology Proposition 4 When M is a Poisson manifold with bracket {, }, the map [ C (M) Der A C (M A,A) ],f [ad(f)] A is a representation of C (M) into C (M A,A). We denote d the operator of cohomology associated to this representation. For any p N, Λ p P ois (M A, ) =C p [C (M),C (M A,A)]

7 Cohomology associated to a Poisson structure 311 denotes the C (M A,A)-module of skew-symmetric multilinear forms of degree p from C (M) intoc (M A,A). We have Λ 0 P ois(m A, ) =C (M A,A). We denote Λ P ois (M A, ) = n Λ p P ois (M A, ). p=0 Thus, for Ω Λ p P ois (M A, ) and f 1,..., f p+1 C (M), we have p+1 dω(f 1,..., f p+1 )= ( 1) i [ad(fi )] A [Ω(f 1,..., f i,..., f p+1 )] i=1 + 1 i<j p+1 ( 1) i+j Ω({f i,f j },f 1,..., f i,..., f j,..., f p+1 ) where f i means that the term f i is omitted. When η Λ p P ois (M), then η A : C (M)... C (M) C (M A,A), (f 1,..., f p ) [η(f 1,..., f p )] A is skew-symmetric multilinear forms of degree p from C (M) intoc (M A,A) i.e η A Λ p P ois (M A, ). Thus Proposition 5 For any η Λ p P ois (M), we have dη A =(dη) A.

8 312 Vann Borhen Nkou and Basile Guy Richard Bossoto Proof. For any f 1,..., f p+1 C (M), we have p+1 ( dη A )(f 1,..., f p+1 )= ( 1) i [ad(fi )] (η A A (f 1,..., f ) i,..., f p+1 ) That ends the proof. i=1 + 1 i<j p+1 ( 1) i+j η A ( {f i,f j },f 1,..., f i,..., f j,..., f p+1 ) p+1 = ( 1) i [ad(fi )] (η(f A 1,..., f i,..., f p+1 ) i=1 + 1 i<j p+1 ) A ( 1) i+j [η({f i,f j },f 1,..., f i,..., f j,..., f p+1 )] A p+1 = ( 1) i {f i,η(f 1,..., f i,..., f p+1 )} A i=1 + 1 i<j p+1 ( 1) i+j [η({f i,f j },f 1,..., f i,..., f j,..., f p+1 )] A =[(dη)(f 1,f 2,..., f p+1 )] A. Corollary 6 The 1-form η A is d-closed i.e ( dη A =0), if and only if dη =0. In particular when η is a derivation of Poisson algebra C (M). Proof. Indeed, for p = 1, we have ( dη A )(f,g) = [ad(f)] A [η A (g)] [ad(g)] A [η A (f)] η A ({f,g}) = {f,η(g)} A {g, η(f)} A [η ({f,g})] A =({f,η(g)} {g, η(f)} η ({f,g})) A =[dη(f,g)] A. for any f,g C (M). Thus dη A = 0 if qnd only if dη =0. When η is a derivation of Poisson algebra C (M), we have f,g C (M), i.e η({f,g}) = {η(f),g} + {f,η(g)} = {f,η(g)} {g, η(f)} ( dη A )(f,g) =[dη(f,g)] A =0. That ends the proof.

9 Cohomology associated to a Poisson structure 313 Proposition 7 If η and η both are cohomologous d-closed p-forms then η A and η A both are cohomologous d-closed p-forms. Proof. For any f 1,..., f p C (M) we have then i.e [η A η A ](f 1,..., f p )=η A (f 1,..., f p ) η A (f 1,..., f p ) =[η(f 1,..., f p )] A [η (f 1,..., f p )] A If there exists ν Λ p 1 P ois (M) such that =[η(f 1,..., f p ) η (f 1,..., f p )] A =[(η η )(f 1,..., f p )] A. η η = dν [η A η A ](f 1,..., f p )=[(η η )(f 1,..., f p )] A =[dν(f 1,..., f p )] A = dν A (f 1,..., f p ). η A η A = dν A. The cohomology class of the d-closed p-form η induces the cohomology class of the d-closed p-form η A. Let Z p P ois (M A, ) be the set of d-closed p-forms from C (M)intoC (M A,A) and B p P ois (M A, ) be the set of d-exact p-forms from C (M) intoc (M A,A). We denote H p P ois (M A, ) =Z p P ois (M A, )/B p P ois (M A, ). For p =0, Λ 0 P ois (M A, ) =C (M A,A). It is obvious that H 0 (M A, ) is the center of C (M A,A) i.e the set { } φ (M A,A); [ad(f)]a (φ) = 0 for every f C (M). For p = 1, we have H 1 P ois(m A, ) =0.

10 314 Vann Borhen Nkou and Basile Guy Richard Bossoto 4 da -Poisson cohomology The map C (M A,A) Der A [C (M A,A)],ϕ τ ϕ, is a representation of C (M A,A)intoC (M A,A). We denote d A the cohomology operator associated to this representation. For any p N, Λ p P ois (M A, A ) = C p [C (M A,A),C (M A,A)]denotes the C (M A,A)-module of skew-symmetric multilinear forms of degree p on C (M A,A) into C (M A,A). We have Λ 0 P ois (M A, A )=C (M A,A). We denote Λ P ois (M A, A )= n Λ p P ois (M A, A ). p=0 For Ω Λ p P ois (M A, A ) and ϕ 1,ϕ 2,..., ϕ p+1 C (M A,A), we have p+1 d A Ω(ϕ 1,..., ϕ p+1 )= ( 1) i 1 τ ϕi [Ω(ϕ 1,..., ϕ i,..., ϕ p+1 ] i=1 + 1 i<j p+1 ( 1) i+j Ω({ϕ i,ϕ j } A,ϕ 1,..., ϕ i,..., ϕ j,..., ϕ p+1 ) i.e p+1 d A Ω(ϕ 1,ϕ 2,..., ϕ p+1 )= ( 1) i 1 {ϕ i, Ω(ϕ 1,..., ϕ i,..., ϕ p+1 } A i=1 + 1 i<j p+1 ( 1) i+j Ω({ϕ i,ϕ j } A,ϕ 1,..., ϕ i,..., ϕ j,..., ϕ p+1 ). For p =1, we have d A Ω(ϕ, ψ) ={Ω(ϕ),ψ} A + {ϕ, Ω(ψ)} A Ω({ϕ, ψ} A ) for any ϕ, ψ C (M A,A). Thus Corollary 8 The 1-form Ω is d A -closed i.e d A Ω=0if, and only if, Ω({ϕ, ψ} A = {Ω(ϕ),ψ} A + {ϕ, Ω(ψ)} A i.e Ω is a derivation of the algebra C (M A,A).

11 Cohomology associated to a Poisson structure 315 Let Z p P ois (M A, A ) be the set of d A -closed p-forms from C (M A,A)into C (M A,A) and B p P ois (M A, A ) be the set of d A -exact p-forms from C (M) into C (M A,A). We denote H p P ois (M A, A )=Z p P ois (M A, A ) /B p P ois (M A, A ). For p =0, Λ 0 P ois (M A, A )=C (M A,A). It is obvious that H 0 (M A, A )is the center of C (M A,A) i.e the set { ϕ C (M A,A); {ϕ, φ} A = 0 for every φ C (M A,A) }. For p = 1, we have H 1 P ois (M A, A )=0. Acknowledgements: The first author thanks Deutscher Akademischer Austauschdientst (DAAD) for their financial support. References [1] B. G. R. Bossoto, E. Okassa, Champs de vecteurs et formes différentielles sur une variété de points proches, Archivum Mathematicum (Brno), 44(2008) [2] B. G. R. Bossoto, Structures de Jacobi sur une variété des points proches, Math. Vesnik. 62, 2 (2010), [3] B. G. R. Bossoto, E. Okassa, A-poisson structures on Weil bundles, Int., J. Contemp. Math. Sciences, Vol. 7, 2012, n 16, [4] I. Kolar, P.W. Michor, and J. Slovak, Natural operations in differential geometry. Springer, 1993, 434 p. [5] A. Lichnerowicz, Les variétés de Poisson et leurs algébres de Lie associées, J. Diff. Geom., 12 (1977), [6] A. Morimoto, Prolongation of connections to bundles of infinitely near points, J. Diff. Geom, t.11(1976), [7] E. Okassa, Prolongement des champs de vecteurs à des variétés de points proches, Annales Faculté des sciences de Toulouse, Vol. VIII, n 3, ,

12 316 Vann Borhen Nkou and Basile Guy Richard Bossoto [8] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Math.118, Birkhäuser Verlag, Basel, [9] V. V. Shurygin, Smooth manifolds over local algebras and Weil bundles, J. Math. Sci., 108 (2) (2002), [10] A. Weil, Théorie des points proches sur les variétés différentiables, Colloq. Géom. Diff. Strasbourg (1953), Received: August 15, 2013