1. Introduction
A 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

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
![]()
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
![]()
such that
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for any
[9] . Moreover the map
![]()
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
![]()
the unique
-skew-symmetric multilinear map such that
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i.e.
induces a derivation
![]()
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
,
![]()
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 Bundles
When
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
![]()
such that for any
and
in
,
![]()
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 Bundles
Proposition 3. For any
and for any
, we have
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Proof. If
, then there exists
, such that
. Thus,
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3.1. Lie Derivative
The 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
![]()
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
![]()
is a Poisson vector field if and only if
![]()
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
![]()
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.
![]()
Thus,
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3.2. Example
When
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,
![]()
![]()
where
for
. For
,
![]()
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As
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we have
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