Journal of Generalized Lie Theory and Applications Modules Over Color Hom-Poisson Algebras

In this paper we introduce color Hom-Poisson algebras and show that every color Hom-associative algebra has a non-commutative Hom-Poisson algebra structure in which the Hom-Poisson bracket is the commutator bracket. Then we show that color Poisson algebras (respectively morphism of color Poisson algebras) turn to color Hom-Poisson algebras (respectively morphism of Color Hom-Poisson algebras) by twisting the color Poisson structure. Next we prove that modules over color Hom–associative algebras A extend to modules over the color Hom-Lie algebras L(A), where L(A) is the color Hom-Lie algebra associated to the color Hom-associative algebra A. Moreover, by twisting a color Hom-Poisson module structure map by a color Hom-Poisson algebra endomorphism, we get another one.


Introduction
Color Hom-Poisson algebras are generalizations of Hom-Poisson algebras introducedin [1], where they emerged naturally in the study of 1-parameter formaldeformations of commutative Hom-associative algebras. Color Hom-Poisson algebrasgeneralize, on the one hand, color Hom-associative [2,3] and color Hom-Lie algebras [2,3] which have been recently investigated by various authors. Onthe other hand, they generalize Hom-Lie superalgebras [4]. These structures arewell-known to physicists and to mathematicians studying differential geometry and homotopy theory. The cohomology theory of Lie superalgebras [5] has been generalizedto the cohomology of Hom-Lie superalgebras in [6]. A cohomology of colorLie algebras was introduced and investigated in [7], and the representations ofcolor Lie algebras were explicitly described in [8]. Modules over Poisson algebras receive various definitions [9,10] we will use theone introduced in [9]. The aim of this paper is to study color Hom-Poisson algebras and modules over color Hom-Poisson algebras. The paper is organized as follows. In section 4, we recall some basic notions related to color Hom-associative algebras and color Hom-Lie algebras. In section 5, we define color Hom-Poisson algebras and point out that to any color Hom-associative algebra ones can associate a color Hom-Poisson algebra. Next, starting from a color Poisson algebra and Poisson algebra morphism we get another one by twisting the associative product and Lie bracket. In section 6, we introduce modules over color Hom-Poisson algebras and prove that starting from a color Hom-Poisson module we get another one by twisting the module structure map by a Hom-Poisson algebra endomorphism. All vector spaces considered are supposed to be over fields of characteristics different from 2.

Preliminaries
Let G be an abelian group. A vector space V is said to be a G-graded if, there exist a family (V a ) a ∈ G of vector subspaces of V such that Let A' be another G-graded algebra. A morphism : ' f A A → of G-graded algebras is by definition an algebra morphism from A to A' which is, in addition an even mapping.

Remark
When α=Id we recover the classical associative color algebra.
Recall that the Hom-associator, aS A of a Hom-algebra A is defined as : : , Observe that 0 A as ≡ when A is a color-Hom-associative algebra.
An even linear map : For all x, y ∈ A.

Definition
∈ ) a bicharacter, and an even linear map : A A α → such that for any , Where ∫  means cyclic summation.
By the ε skew symmetry 3 of the color Hom-Lie bracket {. , .}, the color Hom-Jacobi identity 5 is equivalent to Remark that a color Lie algebra (A, {.,. } ,ε) is a color Hom-Lie algebra with α=Id Morphism of color Hom-Lie algebras are defined similarly to the Definition 4.3, where the color Hom-associative product is replaced by the color Hom-Lie bracket. Examples of color Hom-Lie algebras are provided in [2,3].
The following lemma connects color Hom-associative algebras to color Hom-Lie algebras.
The following theorem is the color version of ( [11], Proposition 4.6).
We have a similar proof for the color Hom-Poisson bracket.

Proof
The proof follows from Theorem.2.
As in the case of Poisson algebras ( [10,12,13]), the cohomology of color Hom-Poisson algebras is described by the cohomology of the underlying color Hom-Lie algebras ( [3]).

Modules Over Color Hom-Poisson Algebras
Twisting a module structure map by an algebra endomorphism, we get another one as stated in the following Lemma.

Proof
The proof is similar to that of ( [14], Lemma 4.5).