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**Ibrahima Bakayoko ^{*}**

Department de Mathematiques, UJNK/Centre Universitaire de N'Zerekore, BP: 50, N’Zerekore, Guinea

- *Corresponding Author:
- Ibrahima Bakayoko,

Department de Mathematiques,

UJNK/Centre Universitaire de N'Zerekore, BP: 50, N’Zerekore, Guinea,

**E-mail:**[email protected]

**Received date:** August 12, 2013; **Accepted date:** September 30, 2014; **Published date:** October 06, 2014

**Citation:** Bakayoko I (2014) Modules Over Color Hom-Poisson Algebras. J Generalized Lie Theory Appl 8:212. doi:10.4172/1736-4337.1000212

**Copyright:** © 2014 Bakayoko I. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

Color hom-associative algebras; Color hom-Lie algebras; Homomorphism; Formal deformation;Hom-modules; Modules over color Hom-Lie algebras; Modules over color Hom-Poisson algebras

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.

Let G be an abelian group. A vector space V is said to be a G-graded if, there exist a family (Va)a ∈ G of vector subspaces of V such that

An element x ∈ V is said to be homogeneous of degree a ∈ G if x ∈ Va We denote H (V) the set of all homogeneous elements in V.

Let and be two G-graded vector spaces. A linear mapping is said to be homogeneous of degree b if

If,f is homogeneous of degree zero i.e. holds for any a ∈ G then f is said to be even.

An algebra (A, μ) is said to be G-graded if its underlying vector space is G-graded i.e. and if furthermore for all a, b ∈ G.

Let A' be another G-graded algebra. A morphism of G-graded algebras is by definition an algebra morphism from A to A' which is, in addition an even mapping.

**Definition**

Let G be an abelian group. A map is called a skewsymmetric bicharacter on G if the following identities hold,

1. ε(a,b) ε(b,a)=1

2. ε (a,b+c)= ε(a,b) ε(a,c)

3. ε (a+b,c)= ε(a,c) ε(b,c)

a,b,c ∈ G

Remark that

ε(a,0)= ε(0,a)=1, ε(a,a)= ± 1 For all a ∈ G

Where, 0 is the identity of G. If x and y are two homogeneous elements of degree a and b respectively and ε is a skew-symmetric bicharacter, then we shorten the notation by writing ε (x , y) instead of ε (a , b)

**Definition**

A color Hom-associative algebra is a quadruple (A, μ, ε, α) consisting of a G-graded vector space A, an even bilinear map and an even linear map such that

μ (α(x), α (y)) = α (μ(x, y))

μ(α(x),μ(y, z))=μ(μ(x, y),α(z))

If in addition μ(x, y) = ε(x, y ) μ (y, x) the color Hom-associative algebra (A, μ, ε, α) is said to be a ε -commutative color Hom-associative algebra.

**Remark**

When α=Id we recover the classical associative color algebra.

Recall that the Hom-associator, aSA of a Hom-algebra A is defined as : as

Observe that when A is a color-Hom-associative algebra.

**Definition**

Let (A, μ, ε, α) and (A' ,μ', ε', α') be two color Hom-associative algebras. An even linear map is said to be a morphism of color Hom-associative algebras if and

f(μ(x, y))=μ' (f (x),f (y))

For all x, y ∈ A.

**Lemma**

([17]) Let (A, μ, ε,) be a color associative algebra and α be an even algebra endomorphism. Then (A, μα, ε, α) whereis a color Hom-associative algebra. Moreover, suppose that (A', μ', ε) be another color associative algebra and be an even algebra endomorphism such that then also a morphism of color Hom-associative algebras.

**Definition**

([17]) A color Hom-Lie algebra is a quadruple (A, {.,.},ε, α) consisting of a G-graded vector space A, an even bilinear map

(i.e for all )a bicharacter, and an even linear map such that for any we have

(26) (ε -Skew-symmetry)

(27) (Multiplicativity)

(28) (ε -Hom-Jacobi identity)

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

(30)

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.

**Lemma**

([17]) Let (A, μ, ε, α) be a color Hom-associative algebra.

Then is a color Hom-Lie algebra, denoted by L (A).

**Definition**

A color Hom-Poisson algebra consists of a G-graded vector space A, a multiplication an even bilinear bracket and an even linear map such that

1. (A, μ, ε, α) is a color Hom-associative algebra,

2. (A, {.,.},ε, α) is a color Hom-Lie algebra,

3. the color Hom-Leibniz identity is satisfied i.e.

(35)

For any

If in addition μ is ε commutative, the color Hom-Poisson algebra (A, {.,.},ε, α) is said to be a ε commutative color Hom-Poisson algebra.

The condition 7 expresses the compatibility between the color Hom-associative product μ and the color Hom-Lie bracket {.,.} it can be written equivalently

(37)

**Remark**

We recover Poisson algebras ([6, 5]) when α = Id and ε ≡1

We need the following lemma in Proposition 6.1.

**Lemma**

If (A, μ {.,.} ,ε, α) is a ε commutative color Hom-Poisson algebra, then(A, - μ - {.,.},ε, α) is also a ε commutative color Hom-Poisson algebra.

The following theorem is the color version of ([11], Proposition 4.6).

**Theorem**

Let (A, μ, ε, α) be a color Hom-associative algebra.

Then ) is a color Hom-Poisson algebra.

**Proof:**

According to Lemma 4.2, it remains to prove the color Hom- Leibniz identity 7. For any

This finishes the proof.

**Corollary**

Let (A, μ {.,.}, ε, α ) be a color associative algebra and α an even color algebra endomorphism. Then where is a color Hom-Poisson algebra.

**Proof**

The proof follows from Lemma 4.1 and Theorem 3.1.

**Lemma**

Let (A, μ {.,.} ,ε) be a color Poisson algebra and α be an even color Poisson algebra endomorphism. Then is a color Hom Poisson algebra.

**Proof**

By Lemma 4.1 and ([3] Example1.2), we only need to prove the color Hom-Leibniz identity. For any

This completes the proof.

**Theorem**

Let (A, μ {.,.} ,ε, α) be a color Hom-Poisson algebra and be an even color Poisson algebra endomorphism. Then, is a color Hom-Poisson algebra.

Moreover, suppose that (A', μ', {.,.}' ε ) is a color Poisson algebra and is an even color Poisson algebra endomorphism. If is a color Poisson algebra morphism that satisfies then

is a color Hom-Poisson algebra homomorphism.

Proof

It is straightforward to show that is a color Hom associative algebra and is a color Hom-Lie algebra ([3] Theorem1.1). The proof of the color Hom-Leibniz identity is similar to that of Lemma 5.2.For the second assertion, we have

We have a similar proof for the color Hom-Poisson bracket.

**Corollary 5.2**

Let (A, μ {.,.} ,ε, α )be a color Hom-Poisson algebra. Then

is a color Hom-Poisson algebra for each integer n ≥ 0. We finish this section by studying deformations by composition of color HomPoisson algebras.

**Definition 5.2**

Let (A, μ {.,.} ,ε, α )be a color Hom-Poisson algebra. A one parameter formal deformation of A is given by K[[t]]-bilinear maps and of the form and where each and {.,.}i are K-bilinear maps and (extended to K[[t]]-bilinear maps), and such that for all following conditions be satisfied.

(67)

(68)

(69)

(70)

(71)

(72)

The deformation is said to be of order k if and

**Proposition 5.1**

Let (A, μ {.,.} ,ε ) be a color Poisson algebra and t α an even color Poisson algebra endomorphism of the form where are endomorphism of A (as color Poisson algebra), t is a parameter in K and k is an entiger.Let and then is a color Hom-Poisson algebra which is a deformation of the color Poisson algebra (A, μ {.,.} ,ε viewed as a color Hom-Poisson algebra (A, μ {.,.} ,ε, Id ).

**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]).

**Definition 6.1**

Let G be an abelian group. A Hom-module is a pair (M, αM ) in which M is a G-graded vector space and is an even linear map.

**Definition 6.2**

Let (A, ΜA, ε, αA) be a color Hom-associative algebra. An Amodule is a Hom-module (M, αM) together with a bilinear map called structure map, such that

(82)

(83)

(84)

Twisting a module structure map by an algebra endomorphism, we get another one as stated in the following Lemma.

**Lemma 6.1**

Let (A, ΜA, ε, αA ) be a color Hom-associative algebra and M an A-module with structure map Define the map

Then is the structure map of another A-module structure on M.

**Proof**

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

**Definition 6.3**

([3]) Let (L, {.,.},ε ,αL ) be a color Hom-Lie algebra and (M, αM ) a Hom-module. An L-module on M consists of a K-bilinear map

(89)

(90)

(91)

(92)

for any

**Remark 6.1**

When we recover the definition of Lie modules ([15-17]).

The following statement is the Lie analogue of Lemma 6.1.

**Lemma 6.2**

Let (L, {.,.},ε ,α ) be a color Hom-Lie algebra and M an L-module with structure map

(96)

Then is the structure map of another L-module structure on M.

**Proof**

Equalities 19 and 20 are proved as in Lemma 6.1. Now, we prove

21 for For any

Hence the conclusion holds.

The following result shows that A-modules extend to L(A)-modules with samemodule structure map.

**Theorem 6.1**

Let (A, μ, ε, α) be a color Hom-associative algebra and (M,ΑM ) an A-module with structure map μM . Then, M is a L(A)-module with structure map μM.

**Proof**

In fact, it suffices to show the relation 21. For any We have

This establishes the Theorem.

The corollaries below give a large class of examples of L(A)- modules.

**Corollary 6.1**

Let be a color Hom-associative algebra as in

Lemma 4.1 and *(M, αM )* an module with structure map* μM*. Then, M is an L(A)-module with structure map *μM*.

**Proof**

Prove 21. Indeed, for we have

We conclude that M is an L(A)-module with structure map μM

**Corollary 6.2**

Let (A, μ, ε, α) be a color Hom-associative algebra and (M, αM ) an

A-module with structure map μM Put

Then M is an L(A)-module with structure map

**Proof**

We know from Lemma (6.1) that is an A-module structure map. And, for one has

This is similar to the relation 21 for

Now we define modules for color Hom-Poisson algebras.

**Definition 6.4**

Let (A, μ, {.,.}, ε, α) be a ε commutative color Hom-Poisson algebra and (M, αM ) a Hom-module.

A color Hom-Poisson module structure on M consists of two K-bilinear maps and such that

(i) M is an A-module and an L-module,

(ii) And for any

When

We recover the definition of modules over Poisson algebras ([9]).

**Example 6.1**

(i) Any module over a ε commutative color Hom-associative algebra (resp. color Hom-Lie algebra) can be seen as a module over a ε commutative color Hom-Poisson algebra with the trivial color Hom- Lie bracket (resp. trivial color Hom-associative product).

(ii) Any ε commutative color Hom-Poisson algebra is a module over itself.

**Example 6.2**

Let (V, μV, λV, αV ) and (W, μW, λW, αW ) be two modules over the ε Commutative color Hom-Poisson algebra (A, μ, {.,.}, ε, α)

Then the direct product is a module over A with structure maps and

Defined by

for any

**Proposition 6.1**

If (M, μM, λM, αM,) is a module over the ε commutative color Hom-Poisson algebra (A, μ, {.,.}, ε, α) then (M, -μM, -λM, αM,) is also a module

Over the ε commutative color Hom-Poisson algebra (A, -μ, -{.,.}, ε, α)

**Proof**

The proof comes from Definition 6.4 and Lemma 5.1.

**Theorem 6.2**

Let (A, μ, {.,.}, ε, α) be A ε commutative color Hom-Poisson algebra and (M, μM, λM, αM,) color Hom-Poisson module. Then

(125)

(126)

Define another color Hom-Poisson module structure on M.

**Proof**

We know that is a structure of another A-module structure map on M (Lemma 6.1) and is a structure of another L-module structure map on M (Lemma 6.2). Show relations 23 and 24 for andFor all

And, for

Hence equations 23 and 24 hold for and This completes the proof.

**Corollary 6.3**

Let (A,μ,{.,.},ε) be A ε commutative color Poisson algebra and (M, ΜM, λM, αM,) A module over the color Hom-Poisson algebra Then , define another color Hom- Poisson module structure on M.

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