http://dx.doi.org/10.4236/apm.2015.513069
Poisson Vector Fields on Weil Bundles
Norbert Mahoungou Moukala
1, Basile Guy Richard Bossoto
1,21Faculty of Sciences and Technology, Marien NGOUABI University, Brazzaville, Congo 2Institut de Recherche en Sciences Exactes et Naturelles (IRSEN), Brazzaville, Congo
Received 2 October 2015; accepted 6 November 2015; published 9 November 2015
Copyright © 2015 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract
In this paper, M is a smooth manifold of finite dimension n, A is a local algebra and MA is the
asso-ciated Weil bundle. We study Poisson vector fields on MA and we prove that all globally
hamilto-nian vector fields on MA are Poisson vector fields.
Keywords
Weil Algebra, Weil Bundle, Poisson Manifold, Lie Derivative, Poisson 2-Form
1. Introduction
A Weil algebra or local algebra (in the sense of André Weil) [1], is a finite dimensional, associative, commuta-tive and unitary algebra A over in which there exists a unique maximum ideal m of codimension 1. In his case, the factor space A m is one-dimensional and is identified with the algebra of real numbers . Thus
A= ⊕ m and is identified with ⋅1A, where 1A is the unit of A.
In what follows we denote by A a Weil algebra, M a smooth manifold, C∞
( )
M the algebra of smooth func-tions on M.A near point of x∈M of kind A is a homomorphism of algebras
( )
:C M A
ξ
∞ →such that for any f ∈C∞
( )
M , ξ( )
f − f x( )
∈m.We denote by A
x
M the set of near points of x of kind A and A A x x M
M M
∈
=
the set of near points on M of kind A. The set MA is a smooth manifold of dimension dimM×dimA and called manifold of infinitely near points on M of kind A [1]-[3], or simply the Weil bundle [4] [5].If f M: → is a smooth function, then the map
( ) ( )
(
)
( )
: ,
A A A A
is differentiable of class C∞ [4] [6]. The set, C∞
(
MA,A)
of smooth functions on AM with values on A, is a commutative algebra over A with unit and the map
( )
(
A,)
, AC∞ M →C∞ M A f f
is an injective homomorphism of algebras. Then, we have:
(
f +g)
A= fA+gA;(
λ⋅f)
A= ⋅λ fA;(
f g⋅)
A= fA⋅gA.We denote
( )
AM
X , the set of vector fields on MA and DerA C
(
MA,A)
∞
the set of A-linear maps
(
)
(
)
: A, A,
X C∞ M A →C∞ M A
such that
(
)
( )
( )
, for any ,(
A,)
.X ϕ ψ⋅ =X ϕ ψ ϕ⋅ + ⋅X ψ ϕ ψ∈C∞ M A
Thus [4],
( )
MA DerA C(
MA,A)
.∞
=
X
If
( )
( )
:C M C M
θ
∞ → ∞is a vector field on M, then there exists one and only one A-linear derivation
(
)
(
)
: , ,
A C MA A C MA A
θ ∞ → ∞
called prolongation of the vector field θ [4] [6], such that
( )
( )
A, for any( )
.A A
f f f C M
θ =θ ∈ ∞
Let Ω C∞
( )
M be the C∞( )
M -module of Kälher differentials of C∞( )
M and( )
( )
( ) ( ): , 1 1
M C M C M f f C M C M f
δ
∞ → Ω ∞ ⊗ ∞ − ∞ ⊗the canonical derivation which the image of δM generates the C∞
( )
M -module Ω C∞( )
M i.e. for( )
x∈ Ω C∞ M ,
( )
:
,
i M i i I finite
x f δ g
∈
=
∑
⋅with f gi, i C
( )
M∞
∈ for any i∈I [7] et [8]. We denote A C
(
MA,A)
∞
Ω , the C∞
(
MA,A)
-module of Kälher differentials of C∞(
MA,A)
which are A-linear. In this case, for ϕ∈C∞(
MA,A)
, we denote(
,)
(
,)
1C MAA 1C MAA
ϕ⊗ ∞ − ∞ ⊗ϕ, the class of
(
,)
(
,)
1C MAA 1C MAA
ϕ
⊗ ∞ − ∞ ⊗ϕ
in C∞(
MA,A)
.The map
( )
(
A,)
,( )
A A 1(
A,)
1(
A,)
AA M C M A C M A
C∞ M → Ω C∞ M A f δ f = f ⊗ ∞ − ∞ ⊗f
is a derivation and there exists a unique A-linear derivation
(
)
(
)
: , ,
A
A A A
A
M C M A C M A
δ
∞ ∞ → Ω
such that
( )
( )
A
A A A
M
M f f
δ
= δ
( )
A(
A,)
, AC∞ M C∞ M A x x
Ω → Ω
is an injective homomorphism of -modules. Thus, the pair
(
(
,)
, A)
A A A C M A δM
∞
Ω satisfies the following
universal property: for every C∞
(
MA,A)
-module E and every A-derivation(
)
:C∞ MA,A E,
Φ →
there exists a unique C∞
(
MA,A)
-linear map(
)
: A C MA,A E
∞
Φ Ω →
such that
. A
A M
δ
Φ = Φ
In other words, there exists a unique Φ which makes the following diagram commutative
(
)
(
)
,
,
A A A
A M
A
C M A
C M A E
δ
∞
Φ
∞
Φ
Ω
↑
→
This fact implies the existence of a natural isomorphism of C∞
(
MA,A)
-modules(
A,)
(
(
,)
,)
(
,)
, , A.A A A
A A M
C M A
Hom ∞ Ω C∞ M A E →Der C∞ M A Eψ ψ δ
In particular, if E=C∞
(
MA,A)
, we have(
A,)
(
A,)
( )
A .A C M A DerA C M A M
∗
∞ ∞
Ω =X
For any p∈, Λ Ωp
(
AC∞(
MA,A)
)
=Lpsks(
ΩAC∞(
MA,A)
,C∞(
MA,A)
)
denotes the C∞(
MA,A)
- module of skew-symmetric multilinear forms of degree p from A C(
MA,A)
∞
Ω into C∞
(
MA,A)
and(
)
(
A,)
p(
(
A,)
)
A A
p
C∞ M A C∞ M A
∈
Λ Ω =
⊕
Λ Ω
the exterior C∞
(
MA,A)
-algebra of(
)
,
A A C M A
∞
Ω called algebra of Kähler forms on C∞
(
MA,A)
.(
)
(
)
(
)
0 A, A, ,
A C M A C M A
∞ ∞
Λ Ω =
(
)
(
)
(
)
1 A, A, .
A C M A A C M A
∗
∞ ∞
Λ Ω = Ω
If p
(
( )
)
,C M
η ∞
∈ Λ Ω then η is of the form δM
( )
f1 ∧ ∧ δM( )
fp with f f1, 2, ,fp C( )
M∞
∈
. Thus,
the C∞
(
MA,A)
-module p(
(
A,)
)
A C M A
∞
Λ Ω is generated by elements of the form
( )
1( )
A A
A A A
p
M M
η =δ ϕ ∧ ∧ δ ϕ
with 1 1A, , A
(
A,)
p p
f f C M A
ϕ = ϕ = ∈ ∞ .
Let
( )
(
( )
)
: C M p p C M ,
θ
σ Ω ∞ → Λ Ω ∞
(
)
( )
1( )
1 2 1
1
ˆ
, , , 1 ,
p i
p i i p
i
x x x x x x x
θ
σ −θ
=
⋅⋅⋅ =
∑
− ⋅ ∧ ∧ ∧ ∧ for any x x1, 2,,xp∈ Ω C∞
( )
M and, where( )
( )
: C M C M
θ Ω ∞ → ∞
is a unique C∞
( )
M -linear map such that θ δ M =θ [8]. Then,(
)
(
(
)
)
: , ,
A
p
A A p A
A C M A A C M A
θ
σ Ω ∞ → Λ Ω ∞
is a unique C∞
( )
M -skew-symmetric multilinear map such that(
1 , 2, ,)
(
1, 2, ,)
.A
A A A A A
p p
x x x θ x x x
θ
σ = σ
We denote
(
(
)
)
1(
(
)
)
: , , ,
A
A p A p A
A C M A A C M A
θ
σ Λ Ω ∞ → Λ − Ω ∞
the unique C∞
(
MA,A)
-skew-symmetric multilinear map such that
(
)
(
)
1 2 1 , 2, ,
A A
A A A A A A A A
p p
x x x x x x
θ θ
σ ∧ ∧ ⋅⋅⋅ ∧ =σ ⋅⋅⋅
i.e. A
A
θ
σ induces a derivation
:
(
(
,)
)
(
(
,)
)
A A
A A A
A A
i C M A C M A
θ σθ
∞ ∞
= Λ Ω → Λ Ω
of degree −1 [9].
We recall that a Poisson structure on a smooth manifold M is due to the existence of a bracket
{ }
, M on( )
C∞ M such that the pair
(
( ) { }
, ,)
M
C∞ M is a real Lie algebra such that, for any f∈C∞
( )
M the map( )
:( )
( )
,{ }
, Mad f C∞ M →C∞ M g f g
is a derivation of commutative algebra i.e.
{
f g h, ⋅}
M ={
f g,}
M⋅ + ⋅h g{ }
f h, Mfor f g h, , ∈C∞
( )
M . In this case we say that C∞( )
M 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
( )
( )
( )
:
M C M C M C M
ω Ω ∞ × Ω ∞ → ∞
such that for any f and g in C∞
( )
M ,{
,}
M M( )
, M( )
M
f g = −ω δ f δ g
defines a structure of Lie algebra over C∞
( )
M [8]. In this case, we say that ωM is the Poisson 2-form of the Poisson manifold M and we denote(
M,ωM)
the Poisson manifold of Poisson 2-form ωM.2. Poisson 2-Form on Weil Bundles
When(
, ,{ }
)
M
M is a Poisson manifold, the map
( )
( )
( )
: ,
ad C∞ M →DerC∞ M f ad f
such that
( ) ( ) {
,}
M ad f g = f g
for any g∈C∞
( )
M , is a derivation. Thus, there exists a derivation(
)
(
)
: , ,
A A A
A
such that
( )
( )
A.A A
ad f = ad f
Let
A:
(
A,)
(
A,)
A A
ad Ω C∞ M A →Der C∞ M A
be a unique C∞
(
MA,A)
-linear map such that .
A
A A A
M
ad δ =ad
Let us consider the canonical isomorphism
(
)
(
)
: , , ,
A A
A A A
A A
M C M A Der C M A M
σ Ω ∞ ∗→ ∞ Ψ Ψ δ
and let
(
)
(
)
(
)
1
1
: , , ,
A MA
A
ad
A A A A
A A A
M ad C M A Der C M A C M A
σ
σ
−
∗
− Ω ∞ → ∞ → Ω ∞
be the map.
Proposition 1. [9] If
(
M,ωM)
is a Poisson manifold, then the map,(
)
(
)
(
)
: , , ,
A
A A A A
A A
M C M A C M A C M A
ω Ω ∞ × Ω ∞ → ∞
such that for any ,
(
A,)
A
X Y∈ Ω C∞ M A
(
,)
1 ( ) ( )
A A
A A
M X Y M ad X Y
ω = −σ−
is a skew-symmetric 2-form on A C
(
MA,A)
∞
Ω such that
(
,)
( )
, ,A
A A A A
M
M x y x y
ω = ω
for any x and y in Ω C∞
( )
M . Moreover,(
, A)
A AM
M ω is a Poisson manifold.
Theorem 2.[9] The manifold MA is a Poisson manifold if and only if there exists a skew-symmetric 2-form
(
)
(
)
(
)
: , , ,
A
A A A A
A A
M C M A C M A C M A
ω Ω ∞ × Ω ∞ → ∞
such that for any ϕ and ψ in C∞
(
MA,A)
,{
,}
A A(
A( )
, A( )
)
A A A
M M M M
ϕ ψ = −ω δ ϕ δ ψ
defines a structure of A-Lie algebra over C∞
(
MA,A)
. Moreover, for any f and g in C∞( )
M ,{
,}
A{
,}
.A A A
M M
f g = f g
In this case, we will say that A
A M
ω is the Poisson 2-form of the A-Poisson manifold MA and we denote
(
, A)
A A M
M ω the A-Poisson manifold of Poisson 2-form A
A M
ω [9].
3. Poisson Vector Field on Weil Bundles
Proposition 3. For any θ∈DerC∞
( )
M and for any η∈ Λ Ω p(
C∞( )
M )
, we have( )
( )
.A
A A
iθ η = iθ η
Proof. If p
(
( )
)
C M
η ∞
∈ Λ Ω , then there exists f f1, 2, ,fp C
( )
M∞
∈
, such that
( )
1( )
M f M fp
( )
(
( )
( )
)
( )
( )
(
)
( )
( )
(
)
( )
( )
(
)
( )
( )
(
)
( )
1
1
1
1
1
, , , ,
.
A A
A
A
A A
M M p
A A
M M p
A A
A
M M p
A
M M p
A
M M p
A
i i f f
i f f
f f
f f
i f f
i
θ θ
θ
θ
θ
θ θ
η δ δ
δ δ
σ δ δ
σ δ δ
δ δ
η
= ∧ ∧
= ∧ ∧
=
=
= ∧ ∧
=
3.1. Lie Derivative
The Lie derivative with respect to D DerA C
(
MA,A)
∞
∈ is the derivation of degree 0
(
)
(
)
(
(
)
)
: , , .
A A
A A A A
D iD δM δM iD A C M A A C M A
∞ ∞
= + Λ Ω → Λ Ω
L
Proposition 4. For any θ ∈X
( )
M , lthe map(
)
(
)
(
(
)
)
: , ,
A
A A
A C M A A C M A
θ
∞ ∞
Λ Ω → Λ Ω
L
is a unique A-linear derivation such that
( )
( )
,A
A A
θ θ η = η
L L
for any η∈ Λ Ω
(
C∞( )
M )
.Proof. For any η∈ Λ Ω
(
C∞( )
M )
, we have( )
( )
( )
( )
(
)
(
( )
)
( )
(
)
(
( )
)
( )
( )
(
)
( )
.A A A A A
A A
A A A A A
M M
A A A
M M
A A
M M
A
M M
A
i i
i i
i i
i i
θ θ θ
θ θ
θ θ
θ θ
θ
η δ η δ η
δ η δ η
δ η δ η
δ η δ η
η
= +
= +
= +
= +
=
L
L
A vector field θ on a Poisson manifold
(
M,ωM)
is called Poisson vector field if the Lie derivative of ωM with respect to θ vanishes i.e. Lθω =M 0. A vector field(
)
(
)
: A, A,
X C∞ M A →C∞ M A
on a A-Poisson manifold of Poisson 2-form A
A M
ω will be said Poisson vector field if LXωMA =0.
Proposition 5. If
(
M,ωM)
is a Poisson manifold, then a vector field( )
( )
:C M C M
θ
∞ → ∞is a Poisson vector field if and only if
(
)
(
)
: , ,
A C MA A C MA A
θ ∞ → ∞
is a Poisson vector field.
Proof. indeed, for any θ ∈X
( )
M ,[
]
.A A
A M
M θ
θ
ω
=ω
Thus, Lθω =M 0 if and only if LθAωMA =0.
Proposition 6. Let
(
, A)
A M
M ω 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 ϕ∈C∞
(
MA,A)
such that X =ad( )
ϕi.e. X is the interior derivation of the Poisson A-algebra C∞
(
MA,A)
[6]. For any ψ and φ∈C∞(
MA,A)
,( )
( )
(
)
(
( ))
(
( )
( )
)
( )
(
( )
( )
)
(
( )( )
( )
)
( )
( )( )
(
)
( ) { }
(
)
{
}
( )
( )
{ }
{ }
{
}
{
{
}
}
{
{ }
}
(
{
{ }
}
{
{
}
}
{
{ }
}
)
, ,
, ,
,
, , , , ,
, , , , , , , , , , , , 0.
A A A A A A
A A A A A A
A A A
A A A A A
A A A A
X M M M ad M M M
A A A A
ad
M M M M M M
A A
ad
M M M
A A A
M
M M M M M
ad
ad
ϕ
ϕ
ϕ
ω δ ψ δ φ ω δ ψ δ φ
ϕ ω δ ψ δ φ ω δ ψ δ φ
ω δ ψ δ φ
ϕ ψ φ ω δ ϕ ψ δ φ ω δ ψ δ ϕ φ
ϕ ψ φ ϕ ψ φ ψ ϕ φ ϕ ψ φ φ ϕ ψ ψ φ ϕ
=
= −
−
= − − −
= − + + = − + + =
L L
L
L
Thus, all globally hamiltonian vector fields are Poisson vector fields.
When
(
M,Ω)
is a symplectic manifold, then(
MA,ΩA)
is a symplectic A-manifold [6] [12]. For(
A,)
C M A
ϕ∈ ∞ , we denote X
ϕ the unique vector field on MA, considered as a derivation of C∞
(
MA,A)
into C∞
(
MA,A)
, such that( )
,A A X
i ϕΩ =d
ϕ
where
(
)
(
)
: , ,
A A A
d Λ M A → Λ M A
denotes the operator of cohomology associated with the representation
( )
MA DerA C(
MA,A)
,X X.∞
→
X
When
(
MA,ΩA)
is a symplectic A-manifold, then for any X∈X( )
MA ,(
)
.A A A XΩ =d iXΩ
L
Therefore, all globally hamiltonian vector fields are Poisson vector fields.
Proposition 7. For any ϕ∈C∞
(
MA,A)
and for any Poisson vector field Y, we have ( ), Y .
Y Xϕ X ϕ
=
Proof.
(
)
(
)
(
)
( )
( )
( )
( )
( ), ,
. Y
A A A A
Y X Y X X Y Y X
A A A A A
Y X Y Y
A A A A
Y X
i i i i
i i d d d i d
d i d d Y i
ϕ ϕ ϕ
ϕ
ϕ
ϕ
ϕ ϕ
ϕ ϕ
Ω = Ω = Ω − Ω
= Ω = +
= = = Ω
L L L
L
Thus,
( )
, Y .
Y Xϕ X ϕ
=
3.2. Example
When
1
n i i i
p dx
α
=
=
∑
is a Liouville form, where(
x1,⋅⋅⋅,x pn, 1,⋅⋅⋅,pn)
is a local system of coordinates in the co-tangent bundle T M∗ of M, then (T M∗ ,1
n
i i i
dα dp dx
=
A
α be the unique differential A-form of degree −1 on T M∗ A such that
( )
( )
A.A A
d α = d α
Thus,
( )
( )
( )
( )
1 . n AA A A A A A A
i i
i
d α d α d p d x
=
Ω = = =
∑
ΛTherefore, (T M∗ A, Ω =A dA
( )
αA ) is a symplectic A-manifold.For H∈C∞
(
MA,A)
, let XH be the globally hamiltonian vector field( )
. HA A X
i Ω = −d H
As [13]
, A A i i x x ∂ ∂ = ∂ ∂ we have 1 1 , n n
H i A i A
i i i i
X f g
x p = = ∂ ∂ = ⋅ + ⋅ ∂ ∂
∑
∑
(
)
( )
(
)
( )
( )
( )
1 1 1 1 , , .H H H
n n
A A A A A A A
X X A i X A i
i i i i
n n
A A A A A A
H A i H A i
i i i i
i i d x i d p
x p
X d x X d p
x p
= =
= =
∂ ∂
Ω = Ω + Ω
∂ ∂
∂ ∂
= Ω + Ω
∂ ∂
∑
∑
∑
∑
As( )
( )
(
)
(
( )
( )
)
( )
( )
1 1 , , , , , , , , n nA A A
H A i A A j A A
j j
i j i j i
n n
A A A A A A A A
j k k A A j k k A A
j k j i j k j i
n
A A A A A
j k A k A
j k j i
X f g
x x x p x
f d p d x g d p d x
x x p x
g d p d x d
p x
= =
∂ ∂ ∂ ∂ ∂
Ω ∂ = ⋅Ω ∂ ∂ + ⋅Ω ∂ ∂
∂ ∂ ∂ ∂
= ⋅ Λ + ⋅ Λ
∂ ∂ ∂ ∂ ∂ ∂ = ⋅ ∂ ∂ −
∑
∑
∑
∑
∑
( )
( )
(
)
, ,A A A
k A k A
i i
n
j kj ki j j k
p d x
x p
g δ δ g
∂ ∂ ∂ ∂ =
∑
⋅ ⋅ = and( )
( )
(
)
( )
( )
( )
1 1 1 , , , , , , , n nA A A
H A i A A j A A
j j
i j i j i
n n
A A A A A
i A A j k k A A
j j i j k j i
n
A A A A A A
j k A k A k A
j k j i i
X f g
p x p p p
f f d p d x
x p x p
f d p d x d p
x p p
= =
=
∂ ∂ ∂ ∂ ∂
Ω ∂ = ⋅Ω ∂ ∂ + ⋅Ω ∂ ∂
∂ ∂ ∂ ∂
= ⋅Ω = ⋅ Λ
∂ ∂ ∂ ∂ ∂ ∂ ∂ = ⋅ ∂ ∂ − ∂
∑
∑
∑
∑
∑
( )
(
)
, . A A k A j nj kj ki j j k
d x x
f δ δ f
( )
( )
( )
( )
1 1
1 1
.
H
n n
A A A A A
X i i i i
i i
n n
A A A A
i i
A A
i i i i
i g d x f d p dH
H H
d x d p
x p
= =
= =
Ω = ⋅ + ⋅ = −
∂ ∂
= −
∂ ∂
∑
∑
∑
∑
Thus,
i A i i A
i H f
p H g
x
∂ =
∂
∂
= −
∂
where ,
(
A,)
i i
f g ∈C∞ U A . An integral curve of XH is a solution the following system of ordinary equation
( )
( )
dd
1, 2, , . d
d
A A i
A A
i A A
i
A A
i
x H
t p
i n
p H
t x
∂
=
∂
∀ =
∂
= −
∂
1 1
n n
H A A A A
i i i i i i
H H
X
p x x p
= =
∂ ∂ ∂ ∂
= −
∂ ∂ ∂ ∂
∑
∑
When
(
x x1, 2,,x2n)
is a local system of coordinates corresponding at a chart U of M,1
.
n
i i n U
i
dx dx+ =
Ω =
∑
ΛThus,
( )
( )
=1
= ,
A n
A A A A A
i i n
U i
d x d x+
Ω
∑
Λ1 1
A
A A
n n
i i n U
i i i i n
X f f
x + x
= = +
∂ ∂
= +
∂ ∂
∑
∑
where fi,fi n+ ∈C∞
(
UA,A)
for i=1, 2,,n. For ϕ∈C∞(
MA,A)
,1 1
.
n n
X A A A A
i xn i xi i xi xn i
ϕ
ϕ ϕ
= + = +
∂ ∂ ∂ ∂
= −
∂ ∂ ∂ ∂
∑
∑
( )
( )
1 1
.
n n
A A A A A
X A i n i
i i i n i
i d x d x
x x
ϕ
ϕ ϕ
+
= = +
∂ ∂
Ω = − +
∂ ∂
∑
∑
As
{
ϕ ψ,}
ΩA =Xϕ( )
ψ ,we have
{
}
1 1
, A .
n n
A A A A
i xn i xi i xi xn i
ϕ ψ ϕ ψ
ϕ ψ Ω
= + = +
∂ ∂ ∂ ∂
= −
∂ ∂ ∂ ∂
∑
∑
References
[1] Weil, A. (1953) Théorie des points proches sur les variétés différentiables. Colloq. Géom. Diff. Strasbourg, 111-117. [2] Morimoto, A. (1976) Prolongation of Connections to Bundles of Infinitely Near Points. Journal of Differential
[3] Okassa, E. (1986-1987) Prolongement des champs de vecteurs à des variétés des points prohes. Annales de la Faculté des Sciences de Toulouse, 3, 346-366.
[4] Bossoto, B.G.R. and Okassa, E. (2008) Champs de vecteurs et formes différentielles sur une variété des points proches.
Archivum Mathematicum, Tomus, 44, 159-171.
[5] Kolár, P., Michor, P.W. and Slovak, J. (1993) Natural Operations in Differential Geometry. Springer-Verlag, Berlin. http://dx.doi.org/10.1007/978-3-662-02950-3
[6] Moukala Mahoungou, N. and Bossoto, B.G.R. (2015) Hamiltonian Vector Fields on Weil Bundles. Journal of Mathe-matics Research, 7, 141-148. http://dx.doi.org/10.5539/jmr.v7n3p141
[7] Laurent-Gengoux, C., Pichereau, A. and Vanhaecke, P. (2013) Poisson Structures. Grundlehren der mathematischen Wissenschaften, 347. www.springer.com/series/138
[8] Okassa, E. (2007) Algèbres de Jacobi et Algèbres de Lie-Rinehart-Jacobi. Journal of Pure and Applied Algebra, 208, 1071-1089. http://dx.doi.org/10.1016/j.jpaa.2006.05.013
[9] Moukala Mahoungou, N. and Bossoto, B. G.R. Prolongation of Poisson 2-Form on Weil Bundles.
[10] Lichnerowicz, A. (1977) Les variétés de Poisson et leurs algèbres de Lie associées. Journal of Differential Geometry, 12, 253-300.
[11] Vaisman, I. (1994) Lectures on the Geometry of Poisson Manifolds. Progress in Mathematics 118, Birkhäuser Verlag, Basel. http://dx.doi.org/10.1007/978-3-0348-8495-2
[12] Bossoto, B.G.R. and Okassa, E. (2012) A-Poisson Structures on Weil Bundles. International Journal of Contemporary Mathematical Sciences, 7, 785-803.
