Abstract

In order to construct a Poisson cohomology complex in the quasi-Poisson context, we establish an isomorphism between interesting Poisson cohomology groups for a quasi-Poisson algebra and Poisson cohomology groups for Poisson algebra coming from the Jacobiator of the quasi-Poisson algebra.

1. Introduction

The Poisson structures were first introduced and discussed in the not so well know paper by S. Lie in 1875 ¹, whose use the name of function groups. The classical Poisson bracket

defined on the algebra of smooth functions on plays a fundamental role in the analytical mechanics. It was discovered by D. Poisson in 1809. The Poisson bracket (1) is derived from a symplectic structure on and it appears as one of the main ingredients of symplectic geometry. The basic properties of the bracket (1) are that it yields the structure of a Lie algebra on the space of functions and it has a natural compatibility with the usual associative product of functions. These facts are of algebraic nature and it is natural to define an abstract notion of a Poisson algebra. Following A. Vinogradov and I. Krasil’shchik in ², J. Braconnier (in ³) has developed the algebraic version of Poisson geometry.

One of the most important notion related to the Poisson geometry is Poisson cohomology which was introduced by A. Lichnerowicz (in ⁴) and in algebraic setting by I. Krasil’shchik (in ⁵). Unlike the De Rham cohomology, Poisson cohomology spaces are almost irrelevant to the topology of the manifold and moreover they have bad functorial properties. They are very large and their actual computation is both more complicated and less significant than in the case of the De Rham cohomology. However they are very interesting because they allow us to describe various results concerning Poisson structures in particular one important result about the geometric quantization of the manifold.

According to ⁶, a Poisson algebra is a commutative associative algebra over carrying a Lie algebra bracket for which each adjoint operator is a derivation of the associative algebra structure. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Lie-Poisson groups are a special case. There are also non commutative Poisson algebras ^{7, 8, 9, 10}, but we will not treat them in this paper. Of course, one can replaceby another bracket whose the Jacobi identity is not always verified. We will call it a quasi-Poisson structure.

The quasi-Poisson structures was introduced by Alekseev, Yvette Kosmann-Schwarzbach and Meinrenken in ^{11, 12}. It appeared as a finite-dimensional alternative to infinite-dimensional constructions of Poisson structures on moduli spaces, proposed in particular by Huebschmann ¹³, Goldman ^{14, 15}, Jerrey and Weitsman ¹⁶. According to Vaisman in ¹⁷, Poisson cohomology plays an important role in obstruction to quantification. Example of Poisson cohomolgy of Poisson algebras are given in ¹⁸ and ¹⁹. A usefull reference for Poisson cohomology of Poisson algebras is ²⁰. However, the construction of a Poisson structure often requires a choice of r-matrices, even if the bi-derivation (deduced from the Jacobi quasi-identity) obtained in the quotient does not seem to depend on this choice. Quasi-Poisson structures then appear as a more natural technique for constructing Poisson structures.

They have indeed good reduction properties, allowing to obtain a Poisson structure when we proceed to the quotient. For most of this paper, will be the algebra of smooth functions on a manifold in which case the bracket is called a Poisson structure on and is called a Poisson manifold. The derivations are represented by vector fields, which are called hamiltonian vector fields.

Since the bracket of functions on a Poisson manifold is a derivation in each argument, it depends only on the first derivativesof and and hence it can be written in the form

(2)

where is a field of skew-symmetric bilinear forms on , i.e., a bivector field.

We call the Poisson tensor. The Jacobi identity for the bracket implies that satisfies an integrability condition which is a quadratic first-order (semilinear) partial differential equation in local coordinates and has the invariant form where the bracket here is the Schouten-Nijenhuis bracket on multivector fields (see ²¹).

Lichnerowicz (in ⁴) observed that the operation of Schouten bracket with a Poisson tensor is a differential on multivector fields, and he began the study of the resulting cohomology theory for Poisson manifolds. In particular, he showed that the map from differential forms to multivector fields determined by is a morphism from the de Rham complex to the Poisson complex.

According to ⁶, in the symplectic case this map is an isomorphism, but the Poisson cohomology spaces are in general quite different from the de Rham cohomology. Then consists of the functions which Poisson commute with everything, the so-called Casimir functions on is the space of infinitesimal Poisson automorphisms modulo hamiltonian vector fields. can be interpret as the space of infinitesimal deformations of the Poisson structure modulo trivial deformations, while receives the possible obstructions to extending infinitesimal deformations.

In the following, where . It is clair that where is a Poisson algebra.

Let be an integer, where and let with bracket is associated to the bivector field defined by

In section 2, we establish that is a quasi-Poisson algebra and there exist a biderivation such that

This derivation is given by the bracket of the algebra . The question arises is: there exists an isomorphism between the Poisson cohomology spaces of the quasi-Poisson algebra and those of the Poisson algebra induced by the Jacobi identity? In other words, under what criteria can we obtain an isomomorphism between and Is it related to the cobord? Is it not related to the associated differentials? The same type of problem appears in ²², when Alekseev proposes in 1994, a finite dimensional construction of a Poisson structure on a moduli space by Hamiltonian reduction of a quasi-Poisson biderivation. There are other constructions on moduli space, proposed in particular by Huebschmann ^{23, 24}, Goldman ^{25, 26}, Jerey and Weitsman ²⁷. We do not details these works which are totally independent of the techniques we develop here.

In this paper, we fix a ground field of characteristic zero, which the reader may think of as being or especially in the context of varieties. The Poisson complex considered is described in ²⁰

Definition 1.1 ²⁰ Let be a Poisson algebra. For the space of cochains of the Poisson cohomology complex denoted by is the vector space of skew-symmetric derivations of The Poisson coboundary operator is the graded linear map (of degree 1)

defined, for , where by a skew-symmetric multi-derivation of :

(3)

For all and for

We obtain the Poisson cohomology complex of

The elements of are called Poisson -cocycles, while the elements of are called Poisson coboundaries.

The elements of -th Poisson cohomology space are Poisson modulo Poisson for all and

The graded vector space is called the Poisson cohomology of

In the following, As graded-vector spaces, the Poisson cohomology of the quasi-Poisson algebra will be denoted by

Our main result is the construction of a Poisson cohomology complex on which is isomorphic to

The general notions of a Poisson algebra and of a Poisson cohomology is described in section 2. The Poisson cohomology complex of the cohomology spaces are detailed in section 3 of this article.

2. On Quasi-Poisson Algebras and Poisson Cohomology

In this section, we first specify the notations and recall some classical results that will be useful later. Let be a smooth variety of dimension

We use the following notations: denotes the commutative algebra of on the space of vector fields over i.e. the of the tangent bundle More generally, for any integer , the space of multivectors field of degree

Consider the coordinates system a multivectors field is written:

Let

Let is the space of -differential forms on

A differential form is defined by

Let

Definition 2.1. ²¹ Let be an integer . The Leibniz bracket of order or multi-derivation on is the application

Such that

a) is the alternating -multilinear.

b) verifies Leibniz’s rule.

Let be the space of Leibniz brackets of order on

Proposition 2.1. ²¹ Let be an integer . The application which to a field of multivectors associates the Leibniz bracket of order defined by for all induces a bijection between and .

Proof. (see ²¹).

Definition 2.2. (Jacobiator) For any Leibniz bracket of order 2, we call the Jacobiator of the application

with

Proposition 2.2. ²¹ The Jacobiator is a Leibniz bracket of order

Proof. It is clear that is -multilinear and that if or then

Now,

We have . Therefore is a Leibniz bracket of order

Considering the bivector field

(4)

for any integer where, and are respectively the partial derivatives in and .

Proposition 2.3. is a Poisson quasi-structure on for which the Jacobiator is

(5)

with

Moreover, the Schouten-Nijenhuis bracket that we will now introduce is an extension of the Lie bracket to the whole algebra of multivectors fields as the following theorem shows.

Theorem 2.1. (Schouten-Nijenhuis) Let be a smooth variety. Then there exists on a bracket called the Schouten-Nijenhuis bracket and verifying the following properties:

a) If and then

b) (Graded anti-commutativity) If and then

c) (The graded Leibniz rule) If , and then

d) (The graded Jacobi identity) If , and then

with

Note that if , and , the Schouten-Nijenhuis bracket of and coincides with the Lie bracket and the Schouten-Nijenhuis bracket of and is

Consider the Jacobiator of proposition According to proposition there exists a field of such that

Proposition 2.4.

Proof. By a simple computation, we have and

Since , we deduce that

This completes the proof of the proposition.

Note that implies that

where is the Jacobiator associated to .

Then is a Leibniz bracket of order On a Poisson manifold with the differential is a complex.

In fact, since the property ensures that squares to zero.

The cohomology of is called the Poisson cohomology of

Let be a quasi-Poisson manifold. Then, defines an operator on the space of multivectors. Its square is in general non-vanishing, In the following, we study conditions for which becomes a differential.

Precisely, we are using the Poisson complex described in ²⁰. According to Definition we shall use the following isomorphisms :

With these isomorphisms, we deduce the following proposition.

Proposition 2.1. The following sequence is a Poisson cohomology complex of

(10)

where for in

for all belongs to and where be an integer.

Proof. It follows from definition that it suffices to show that Let in by a simple computation we obtain Thus, is a Poisson cohomology complex of .

Let’s consider now the quasi-Poisson structure on In order to describe the probably Poisson cohomology complex of we will need the following lemma.

Lemma 2.1. For a in the map defined by

is a derivation and the bracket induces a Hamiltonian defined as

with

Proof.

Let It’s clair that is a derivation.

Like any (quasi)-Poisson structure, induces a Hamiltonian defined for all in by which means that

Let be three elements of such that

For that, is defined by

For the sake of simplicity we shall use the following isomorphisms:

With these isomorphisms and considering the following sequence

(11)

where are associated differentials defined by:

for all in , for all in where

where be an integer.

By a simple computation, we deduce that:

Proposition 2.5. is non-vanishing.

We call is a quasi-Poisson cohomology complex of In the following section, we study conditions for which becomes a differential and we construct a sub-algebra of where the Poisson cohomology of is isomorphic to the Poisson cohomology of

3. An Isomorphism between the Quasi-Poisson Cohomology and Poisson Cohomology Associated to the Jacobiator

In this section, we establish an isomorphism between Poisson cohomology groups for the quasi-Poisson algebra and Poisson cohomology of.

Recall the quasi-differential defined in (11) as

Let We have:

with

and

We proved that i.e. is a quasi-differential.

Let’s consider the following partial differential equations:

Let where denotes the solution set of the system of partial differential equations .

Therefore,

Let’s consider the isomorphisms 

Considering the following sequence

with for all belongs to for all in where

where be an integer.

By a simple computation, we obtain the following result.

Proposition 3.1.

is a differential and

Since is a sub-algebra of , the proposition 3.1 completes the proof of the following main result.

Theorem. Let be an integer, and let be the quasi-Poisson algebra defined on with quasi-Poisson structure associated to the bivector field defined by

- The Jacobiator of is equivalent to the Poisson structure .

- As graded- vector spaces, the Poisson cohomology for the quasi-Poisson algebra is isomorphic to Poisson cohomology for the Poisson algebra according to the differential .

This means that there exists an isomorphism between Poisson cohomology groups for a quasi-Poisson algebra and Poisson cohomology groups for Poisson algebra coming from the Jacobiator of the quasi-Poisson algebra. An extension of the work in this context is to explicitly calculate the Poisson cohomology of

Acknowledgements

This work is a part of my Ph.D. at the University of Maroua. I would like to sincerely thank my advisors, Bitjong Ndombol and Joseph Dongho for suggesting to me this interesting problem and for stimulating discussions and the precious hoursthat they spent in proofreading this paper. I especially want to thank Alidou Mohamadou and Elisabeth Ngo Bum for all their support and funding.

References