^{1}

^{*}

^{2}

^{*}

In this paper,
*M* is a smooth manifold of finite dimension n,
*A* is a local algebra and
*M*
^{A} is the associated Weil bundle. We study Poisson vector fields on
*M*
^{A} and we prove that all globally hamiltonian vector fields on
*M*
^{A} are Poisson vector fields.

A Weil algebra or local algebra (in the sense of André Weil) [

In what follows we denote by A a Weil algebra, M a smooth manifold,

A near point of

such that for any

We denote by

kind A. The set

If

is differentiable of class

is an injective homomorphism of algebras. Then, we have:

We denote

such that

Thus [

If

is a vector field on M, then there exists one and only one A-linear derivation

called prolongation of the vector field

Let

the canonical derivation which the image of

with

We denote

The map

is a derivation and there exists a unique A-linear derivation

such that

for any

is an injective homomorphism of

there exists a unique

such that

In other words, there exists a unique

This fact implies the existence of a natural isomorphism of

In particular, if

For any

the exterior

If

the

with

Let

be the

for any

is a unique

is a unique

We denote

the unique

i.e.

of degree −1 [

We recall that a Poisson structure on a smooth manifold M is due to the existence of a bracket

is a derivation of commutative algebra i.e.

for

The manifold M is a Poisson manifold if and only if there exists a skew-symmetric 2-form

such that for any f and g in

defines a structure of Lie algebra over

When

such that

such that

Let

be a unique

Let us consider the canonical isomorphism

and let

be the map.

Proposition 1. [

such that for any

is a skew-symmetric 2-form on

for any x and y in

Theorem 2. [

such that for any

defines a structure of A-Lie algebra over

In this case, we will say that

Proposition 3. For any

Proof. If

The Lie derivative with respect to

Proposition 4. For any

is a unique A-linear derivation such that

for any

Proof. For any

A vector field

on a A-Poisson manifold of Poisson 2-form

Proposition 5. If

is a Poisson vector field if and only if

is a Poisson vector field.

Proof. indeed, for any

Thus,

Proposition 6. Let

Proof. Let X be a globally hamiltonian vector field, then there exists

Thus, all globally hamiltonian vector fields are Poisson vector fields.

When

where

denotes the operator of cohomology associated with the representation

When

Therefore, all globally hamiltonian vector fields are Poisson vector fields.

Proposition 7. For any

Proof.

Thus,

When

Thus,

Therefore, (

For

As [

we have

As

and

As,

Thus,

where

When

Thus,

where

As

we have

Norbert Mahoungou Moukala,Basile Guy Richard Bossoto, (2015) Poisson Vector Fields on Weil Bundles. Advances in Pure Mathematics,05,757-766. doi: 10.4236/apm.2015.513069