1. IntroductionA Weil algebra or local algebra (in the sense of André Weil) [1] , is a finite dimensional, associative, commutative and unitary algebra A over in which there exists a unique maximum ideal of codimension 1. In his case, the factor space is one-dimensional and is identified with the algebra of real numbers. Thus and is identified with, where is the unit of A.
In what follows we denote by A a Weil algebra, M a smooth manifold, the algebra of smooth functions on M.
A near point of of kind A is a homomorphism of algebras
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such that for any,.
We denote by the set of near points of x of kind A and the set of near points on M of
kind A. The set is a smooth manifold of dimension and called manifold of infinitely near points on M of kind A [1] - [3] , or simply the Weil bundle [4] [5] .
If is a smooth function, then the map
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is differentiable of class [4] [6] . The set, of smooth functions on with values on A, is a commutative algebra over A with unit and the map
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is an injective homomorphism of algebras. Then, we have:
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We denote, the set of vector fields on and the set of A-linear maps
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such that
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Thus [4] ,
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If
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is a vector field on M, then there exists one and only one A-linear derivation
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called prolongation of the vector field [4] [6] , such that
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Let be the -module of Kälher differentials of and
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the canonical derivation which the image of generates the -module i.e. for ,
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with for any [7] et [8] .
We denote, the -module of Kälher differentials of which are A-linear. In this case, for, we denote, the class of in.
The map
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is a derivation and there exists a unique A-linear derivation
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such that
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for any [9] . Moreover the map
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is an injective homomorphism of -modules. Thus, the pair satisfies the following universal property: for every -module E and every A-derivation
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there exists a unique -linear map
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such that
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In other words, there exists a unique which makes the following diagram commutative
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This fact implies the existence of a natural isomorphism of -modules
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In particular, if, we have
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For any, denotes the - module of skew-symmetric multilinear forms of degree p from into and
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the exterior -algebra of called algebra of Kähler forms on.
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If then η is of the form with. Thus,
the -module is generated by elements of the form
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with.
Let
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be the -skew-symmetric multilinear map such that
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for any and, where
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is a unique -linear map such that [8] . Then,
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is a unique -skew-symmetric multilinear map such that
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We denote
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the unique -skew-symmetric multilinear map such that
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i.e. induces a derivation
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of degree −1 [9] .
We recall that a Poisson structure on a smooth manifold M is due to the existence of a bracket on such that the pair is a real Lie algebra such that, for any the map
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is a derivation of commutative algebra i.e.
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for. In this case we say that is a Poisson algebra and M is a Poisson manifold [10] [11] .
The manifold M is a Poisson manifold if and only if there exists a skew-symmetric 2-form
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such that for any f and g in,
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defines a structure of Lie algebra over [8] . In this case, we say that is the Poisson 2-form of the Poisson manifold M and we denote the Poisson manifold of Poisson 2-form.
2. Poisson 2-Form on Weil BundlesWhen is a Poisson manifold, the map
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such that for any, is a derivation. Thus, there exists a derivation
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such that
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Let
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be a unique -linear map such that
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Let us consider the canonical isomorphism
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and let
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be the map.
Proposition 1. [9] If is a Poisson manifold, then the map,
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such that for any
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is a skew-symmetric 2-form on such that
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for any x and y in. Moreover, is a Poisson manifold.
Theorem 2. [9] The manifold is a Poisson manifold if and only if there exists a skew-symmetric 2-form
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such that for any and in,
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defines a structure of A-Lie algebra over. Moreover, for any f and g in,
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In this case, we will say that is the Poisson 2-form of the A-Poisson manifold and we denote the A-Poisson manifold of Poisson 2-form [9] .
3. Poisson Vector Field on Weil BundlesProposition 3. For any and for any, we have
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Proof. If, then there exists, such that . Thus,
3.1. Lie DerivativeThe Lie derivative with respect to is the derivation of degree 0
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Proposition 4. For any, lthe map
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is a unique A-linear derivation such that
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for any.
Proof. For any, we have
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A vector field on a Poisson manifold is called Poisson vector field if the Lie derivative of with respect to vanishes i.e.. A vector field
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on a A-Poisson manifold of Poisson 2-form will be said Poisson vector field if.
Proposition 5. If is a Poisson manifold, then a vector field
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is a Poisson vector field if and only if
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is a Poisson vector field.
Proof. indeed, for any,
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Thus, if and only if.
Proposition 6. Let be a A-Poisson manifold. Then, all globally hamiltonian vector fields are Poisson vector fields.
Proof. Let X be a globally hamiltonian vector field, then there exists such that i.e. X is the interior derivation of the Poisson A-algebra [6] . For any and,
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Thus, all globally hamiltonian vector fields are Poisson vector fields.
When is a symplectic manifold, then is a symplectic A-manifold [6] [12] . For
, we denote the unique vector field on, considered as a derivation of into, such that
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where
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denotes the operator of cohomology associated with the representation
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When is a symplectic A-manifold, then for any,
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Therefore, all globally hamiltonian vector fields are Poisson vector fields.
Proposition 7. For any and for any Poisson vector field Y, we have
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Proof.
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Thus,
3.2. ExampleWhen is a Liouville form, where is a local system of coordinates in the cotangent bundle of M, then (,) is a symplectic manifold on [7] . Let be the unique differential A-form of degree −1 on such that
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Thus,
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Therefore, (,) is a symplectic A-manifold.
For, let be the globally hamiltonian vector field
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As [13]
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we have
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As
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and
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As,
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Thus,
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where. An integral curve of is a solution the following system of ordinary equation
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When is a local system of coordinates corresponding at a chart U of M,
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Thus,
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where for. For,
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As
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we have
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