On Theory of logarithmic Poisson Cohomology

We define the notion of logarithmic Poisson structure along a non zero ideal I of an associative, commutative algebra A and prove that each logarithmic Poisson structure induce a skew symmetric 2-form and a Lie-Rinehart structure on the A-module ΩK(logI) of logarithmic Kähler differential. This Lie-Rinehart structure define a representation of the underline Lie algebra. Applying the machinery of Chevaley-Eilenberg and Palais, we define the notion of logarithmic Poisson cohomology which is a measure obstructions of Linear representation of the underline Lie algebra for which the grown ring act by multiplication.


Introduction
The first Poisson bracket on the algebra of smooth functions on R 2n ; was defined by S.D. Poisson in 1809.This bracket plays a fundamental role in the analytical mechanics.One century later that A. Lichnerowicz in (Lichnerowicz, A. (1977)) and A. Weinstein in (Weinstein, A. (1983)) extended it in a large theory now known as the Poisson Geometry.It has been remarked by A. Weinstein that the theory can be traced back to S. Lie in (Lie, S. (1888)).The Poisson bracket (1) is derived from a symplectic structure on R 2n 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.
One of the most important notion related to the Poisson geometry is Poisson cohomology which was introduced by A.
Lichnerowicz in (Lichnerowicz, A. (1977)) and in algebraic setting by I. Krasil'shchik in (Krasil'shchik, I. (1988)).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.Algebraic aspects of this theory were developed by J. Huebschmann in (Huebschmann, J. (2013)) and in the geometrical setting by I. Vaisman in (Vaisman, I. (1991).).
This paper deals with Poisson algebras but Poisson algebras of another kind.More precisely we study the logarithmic Poisson structures.If the Poisson structures draw their origins from symplectic structures, logarithmic Poisson structure are inspired by log symplectic structures which are based on the theory of logarithmic differential forms.The logarithmic differential forms was introduced by P. Deligne in (Deligne, P. (2006)) who defined them in the case of a normal crossings divisor of a given complex manifold.But the theory of logarithmic differential forms along a divisor without necessarily normal crossings was introduced by K. Saito in (Saito, K. (1980)).Explicitly if I is an ideal in a commutative algebra A over a commutative ring R a derivation D of A is called logarithmic along I if D(I) ⊂ I.We denote by Der A (log I) the A-module of derivations of A logarithmic along I.A Poisson structure {., .} on A is called logarithmic2 along I if for all a ∈ A we have {a, .}∈ Der A (log I).In addition suppose that I is generated by {u 1 , ..., u p } ⊂ A and let Ω A be the A-module of Kähler differential.The A-module Ω A (log I) generated by { du 1 u 1 , ..., du p u p } ∪ Ω A is called the module of Kähler differentials logarithmic along I.
J. Huebschmann's program of algebraic construction of the Poisson cohomology can be summarized as follows: Let A be a commutative algebra over a commutative ring R. A Lie-Rinehart algebra on A is an A-module which is an R-Lie algebra acting on A with suitable compatibly conditions.J. Huebschmann observes that each Poisson structure {., .}gives rise to a structure of Lie-Rinehart algebra in the sense of G. Rinehart in (Rinehart, G. S. (1963)) on the Amodule Ω A in natural fashion.But it was proved in (Palais, R. (1961)) that any Lie-Rinehart algebra L on A gives rise to a complex Alt A (L, A) of alternating forms which generalizes the usual De Rham complex of manifold and the usual complex computing Chevalley-Eilenberg in (Chevalley, C., & Eilenberg, S. (1948)) Lie algebra cohomology.Moreover extending earlier work of Hochshild Kostant and Rosenberg in (Hochschild, G., Kostant, B., & Rosenberg, A. (2009)).G.
Rinehart has shown that when L is projective as an A-module the homology of the complex Alt A (L, A) may be identified with Ext * U(A,L) (A, A) over a suitably defined universal algebra U(A, L) of differential operators.But the latter defines a Lie algebra cohomology H * (L, A) of L. So, since Ω A is free A-module, it is projective.Therefore the homology of the complex Alt A (Ω A , A) computing the cohomology of the underlying Lie algebra of the Poisson algebra (A, {., .}).Then the Poisson cohomology of (A, {., .}) is the homology of Alt A (Ω A , A).
It follows from the definition of logarithmic Poisson structure that the image of Hamiltonian map of logarithmic principal Poisson structure is sub-module of Der A (log I).Inspired by this fact we introduce the notion of logarithmic Lie-Rinehart structure.A Lie-Rinehart algebra L on A is said to be logarithmic along an ideal I of A if it acts by logarithmic derivations on A.
If I is an ideal of an associative commutative algebra A over a field of characteristic zero, denoted by Der K (log I) the submodule of Der K (log I) constituted by ν ∈ Der K (log I) such that ν(u) ∈ uA, we prove the following: is an homomorphism of A-modules.Where δ : • For each logarithmic Poisson structure {−, −} : A ⊗ A → A, there exist a unique A-module homomorphism 2. On the Lie-Rinehart Algebra of Logarithmic Principal Differential Form

Logarithmic Derivation and Logarithmic Formal Differential.
In this section, we recall the notion of logarithmic derivation along a non zero ideal of an associative, commutative and unitary algebra A.

Module of Logarithmic Derivations
Let k be a field of characteristic zero and A a k-algebra.Der k (A) the A-module of derivations on A, I a non zero ideal of A and d A/K the universal derivation associated to the A-module of Kähler differentials Ω k (A).It follows from the definition and the fact that A is commutative, that Der K (log Then σ A (ψ) ∈ Der K (A).On the other hand, The map φ : d → d, where d is the unique A-module homomorphism from Where φ denote the inverse of σ A .
we have the following proposition.
Proposition 2.2.σ A : In what follow, we will designate φ the inverse of σ A .

Module of Formals Logarithmic Differentials
In this subsection we define the module of logarithmic Kähler differential.
Let I := I * ∪ {1 A }. We denote I −1 Ω K (A) the localized of Ω K (A).Since Ω K (A) is generated by d(a) and a ∈ A, Definition 2.3.Ω K (log I) is called A-module of logarithmic formals differentials or module of logarithmic Kähler differential along I.
As A-module, Ω K (log I) is not free in general.Indeed, for all is not free.
Follow the K. Saito in (Saito, K. (1980)), when Ω K (log I) is free A-module, I is called a "free ideal" It is proved in ( 12) that for all A-module M, each δ ∈ Der K (A, M) induce a homomorphism of A-modules δ : In this case, we consider the homomorphism δ : We have prove the following lemma.
Lemma 2.4.Let Der K (log is an homomorphism of A-modules.

Logarithmic Poisson Structures
In this section, we introduce the notion of logarithmic Poisson structure and give some of it properties.

Definition and First Properties
Firstly, we recall that a Poisson structure on an algebra A is a skew-symmetric K-bilinear map on A that satisfy the Leibnitz role and Jacobi identity.Proof.Since ad∈Der k (A, Der K (log I)), from universal property of (Ω K (A), d), there exist Proposition 3.5.Each logarithmic Poisson structure along I induce an A-module homomorphism H from Ω K (log I) to Der K (log I), defined by Proof.
For all u ∈ I, , since for all a ∈ A, there exist b ∈ A such that {u, a} = ub.We extended H on Ω K (log I) by linearity.

It follows from this proposition and definition of
We have the following lemma.
Lemma 3.6.Each logarithmic Poisson structure induce a homomorphism of A-modules We deduce that Proposition 3.7.Each logarithmic Poisson structure along I induce a 2-form ω 0 on Ω K (log I).
x i x j {a i ; a j } = 0 Definition 3.9.ω 0 is called Poisson 2-form logarithmic along I. Since We remark that ω0 is equal to the 2-form π {,} defined in (Huebschmann, J. ( 2013)) for an arbitrary Poisson structure {−, −} on A. By a simple computation, we have Theorem 3.10.For all u, v ∈ I * and a, b ∈ A we have the following: From this theorem, it follow that the restriction of ω 0 on

Complex of Logarithmic Differential Form
In this section, we recall the notion of Lie-Rinehart algebra thanks to him we can define the complex of logarithmic differentials forms.

Lie-Rinehart Algebra and Logarithmic Lie-Rinehart Algebra
We conserve the above notations that, a Lie algebra on k is a pair (g; [−, −]), where g is a k-module and [−, −] : g ⊗ g → g k-bilinear skew symmetric map satisfy the Jacobi identity.A Lie-Rinehart algebra is a pair (g, ρ) where g is an A-module and a k-Lie algebra, and ρ : g :→ Der K (A) is a morphism of A-modules and k-Lie algebras, such that for all g, g ′ ∈ g and all a ∈ A, see Rinehart (1963).We can observe that the Lie-Rinehart algebra is the algebraic analogue of a Lie algebroide, it is also known as a Lie pseudo-algebra or a Lie-Cartant pair.
When g is a subset of Der K (A) and ρ : g :→ Der K (A) is the inclusion map, the pair (g, ρ) is a Lie-Rinehart algebra if and only if g is closed under the A-module and k-Lie algebra structures of Der K (A).We will be mainly interested in Lie-Rinehart algebras of this type.An example of this type of Lie-Rinehart algebra is (Der K (log I), i).Which will be referred to Lie-Rinehart structure the map ρ : g → Der K (A) such that (g, ρ) is Lie-Rinehart.Since Der K (log I) is subset of Der K (A) closed under the A-module and k-Lie algebra structures of Der K (A), it is possible for a giving Lie-Rinehart algebra (g, ρ) to verify ρ(g) ⊂ Der K (log I). in this case, we obtain a particular type of Li-Rinehart algebra.More generally, if I is an ideal of A, we have the following definition.It is clear that each log-Lie-Rinehart algebra is a Lie-Rinehart algebra.
In general we can replace in the definition of Lie-Rinehart structure Der K (A) by the A-module of first order differentials operators on some A-module M; see (Dongho, J.(2012)).
If g is a Lie algebra with zero torsion, then each morphism of A-modules ρ : g → Der K (log This end the proof of the following proposition Proposition 4.2.Let g be a Lie algebra such that Ann(g) = 0, the module of annulation of g.Then each morphism of A-modules, ρ : g → Der K (log I) satisfying ( 2) is a logarithmic Lie-Rinehart structure.
Let (g, ρ) be a Lie-Rinehart algebra with associated Lie-bracket [−, −].ρ : g → Der K (A) defines a representation of g by derivation on A, in the case of log-Lie-Rinehart algebra, we have the representation by logarithmic derivation.Using the machinery of Chevalley-Eilenberg and Palais, we define a differential complex (Alt * (g, A), d ρ ), where The associated cohomology is denoted by H * (Alt(g, A), A).This generalized the De Rham cohomology when A is the algebra of smooth functions on a smooth manifold.Indeed, the De Rham cohomology correspond to the case g = Der K (A).Similarly, the case where g = Der X (log D), the O X -module of vector field logarithmic along a reduced divisor D of a complex manifolds X, give the logarithmic of De Rham cohomology.
One of the much important Lie-Rinehart-Closed-Symplectic algebra is the algebra O X of holomorphic map on a logarithmic complex manifold of complex dimension 2n (X, D) with Lie-Rinehart-Closed structure a closed 2-form ω logarithmic along a reduced divisor D of X such that ).In the field of symplectic geometry, such structure are called log symplectic structure and the underline complex manifold is called log symplectic manifold.

Logarithmic Lie Derivative
It is well known that, see (Braconnier, J. (1977)), the map d : ] the exterior A-algebra of the A-module Ω K (log I).d prolonged to a derivation of degree +1, we also defined d; the unique morphism such that The map i δ is a derivation of degree -1.Therefore, ] is a derivation of degree zero.We denote L δ = i δ • d + d • i δ ; to be the logarithmic Lie derivation with respect to δ. L δ have the following properties.
Proof.This is straightforward and left to the reader.
With those properties, we can describe L δ on the generators of Ω K (log I).
Firstly, we shall mark that for all x a ∈ I . This end the proof of the following lemma Immediately, we deduce the following corollary Corollary 5.3.for all a ∈ I * , δ ∈ Der K (log Proof. The following proposition explicite L on generators of Ω K (log I).
Proposition 5.5.For all a ∈ A, u, v ∈ I 2. From proposition 5.1 we deduce that; 3. Changing the role of u and v, we obtain: From this proposition, we deduce the following corollary Proof.This is obvious from proposition 5.1 Corollary 5.7.−dω 0 (ad(u), bd(v)) Proof.This coming from those equalities: We also have: Proof.For all a, b ∈ A, u, v ∈ I * we have: We need the following lemmas to prove the above theorem.Proof.It is straightforward computation.
It is remaining the Jacobian identity to end the proof of the theorem.In this goal, we state the above lemma.
[ , b}dv + b{a, v}du + abd({u, v}) 4. [ On the other hand, as a Poisson structure in the general meaning, {−, −} induce on Ω K (A) a Lie structure.In Theorem 3.8 of (Huebschmann, J. ( 2013)) it is prove that this bracket is defined by [adu, bdv] = a{u, b}dv + b{a, v}du + abd ({u, v}) which is equal to relation 3) of 5.12.We can then conclud that the two brackets coincide on Ω K (A).
In the particular case where a = b = 1, we obtain the following.
The following lemma prove the Theorem on the subset of Ω K (log I) generated by element of the form du u , uI.
Lemma 5.14.For all u, v, w ∈ I * , on a: Proof.Let u, v, w as in the lemma.
we have show in 5.13 that Then Applying lemma [5.10], we obtain: ) .
consequently, Jacobian identity implies, Similarly, we prove that E4 Using those lemmas, we can prove the following proposition that is a part of Theorem 5.9.
Proof.From the above lemmas, we have [ To simplify this expression, we need the following properties of Lie's brackets Lemma 5.17.With the same hypothesis we have.: Proof.Let a 1 ; a 2 ; a 3 ; u 1 ; u 2 ; u 3 as in the lemma.
This equal to zero since {−, −} satisfies the Jacobian identity.
2. Applying Lemma [5.10], we obtain: In the same manner, we prove the rest.
Therefore, from Jacobian identity and relation E 1 we have: We need the following Lemma 5.18.With the same hypothesis, we have: 1. [[ 2. [[ Proof.It follow from lemma 5.10 and above properties of [−, −].
From Lemma 5.10 we have: Applying Leibniz rule on the right hand side of the above equations, we obtain: This end the prove of the following corollary Corollary 5.19. [[ Using the same methods, we prove that 1. [[ Regrouping properly the terms we prove the Jacobian identity.
With all those results we can complete the prove of theorem 5.9 Proof Definition 2.1.A k-derivation logarithmic along I is an element d of Der k (A) such that d(I) ⊂ I.The set of k-derivations logarithmic along I is denoted by Der K (log I).
us denote Der K (log I) the submodule of Der K (log I) constituted by ν ∈ Der K (log I) such that ν(u) ∈ uA.The above construction induce an homomorphism φ : Der K (log I) → Ω K (log I) ∨ δ → δ .
Definition 3.1.A logarithmic Poisson structure along a non zero ideal I of A is a skew-symmetric k-bilinear map {−, −} : A ⊗ A → A that is bi-derivative, satisfy the Jacobi identity and such that {a, u} ∈ uA for all a ∈ A, u ∈ I.It follows from this definition that the image of associated adjoint map ad : A → Der K (A) is a submodule of Der K (log I).Indeed, for all a ∈ A, ad(a) := {a, −} and for all uI, {a, u} ∈ uA ⊂ I. Then ad(a)(I) ⊂ I and ad(A).This end the proof of the following proposition.Proposition 3.2.Let {−, −} : A ⊗ A → A be a logarithmic Poisson structure along I. Then For all a ∈ A, ad(a) ∈ Der K (log I).Corollary 3.3.Let {−, −} : A ⊗ A → A be a logarithmic Poisson structure along I. ad : A → Der K (log I) is homomorphism of Lie algebras and a derivation with values in the A-module Der K (log I).From the universal property of Ω K (A), we deduce Corollary 3.4.For each logarithmic Poisson structure {−, −} : A ⊗ A → A, there exist a unique A-module homomorphism H : Ω K (A) → Der K (log I) such that H • d = ad.
Definition 4.1.A logarithmic Lie-Rinehart algebra along I or shortly log-Lie-Rinehart algebra, is a pair (g, ρ), where ρ : g → Der K (log I) is a morphism of A-modules and k-Lie algebras, satisfying (2).
Since δ ∈ Der K (log I), there exist c ∈ A * such that δ(a) = ac. in subsection 3.2 that each logarithmic Poisson structure induce a map H : Ω K (log I) → Der K (log I). ) for all a, b ∈ I *