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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Modules Over Color Hom-Poisson Algebras

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.

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Abstract

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.

Keywords

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

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 (Va)a ∈ G of vector subspaces of V such that

image

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 imageand imagebe two G-graded vector spaces. A linear mapping imageis said to be homogeneous of degree b if image

If,f is homogeneous of degree zero i.e. image 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. image and if furthermore image for all a, b ∈ G.

Let A' be another G-graded algebra. A morphism image 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 imageis 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 image and an even linear map such image 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 : asimage

Observe that imagewhen 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 imageis said to be a morphism of color Hom-associative algebras if imageand

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) whereimageis a color Hom-associative algebra. Moreover, suppose that (A', μ', ε) be another color associative algebra and image be an even algebra endomorphism such that imagethen image 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

image (i.e image for all image )a bicharacter, and an even linear map imagesuch that for any imagewe have

image(26) (ε -Skew-symmetry)

image (27) (Multiplicativity)

image (28) (ε -Hom-Jacobi identity)

Where imagemeans cyclic summation.

By the ε skew symmetry 3 of the color Hom-Lie bracket {. , .}, the color Hom-Jacobi identity 5 is equivalent to

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

Color Hom-Poisson algebras

Definition

A color Hom-Poisson algebra consists of a G-graded vector space A, a multiplication image an even bilinear bracket imageand an even linear mapimage 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.

image(35)

Forimage 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

image(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 image) is a color Hom-Poisson algebra.

Proof:

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

image

This finishes the proof.

Corollary

Let (A, μ {.,.}, ε, α ) be a color associative algebra and α an even color algebra endomorphism. Then imagewhere image 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 image 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 image

image

This completes the proof.

Theorem

Let (A, μ {.,.} ,ε, α) be a color Hom-Poisson algebra and imagebe an even color Poisson algebra endomorphism. Then,image 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 imageis a color Poisson algebra morphism that satisfies image then

image

is a color Hom-Poisson algebra homomorphism.

Proof

It is straightforward to show that image is a color Hom associative algebra and image 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

image

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

image

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 imageand imageof the form imageand imagewhere each and {.,.}i are K-bilinear maps and (extended to K[[t]]-bilinear maps), and imagesuch that for allimage following conditions be satisfied.

image(67)

image (68)

image(69)

image (70)

image (71)

image(72)

The deformation is said to be of order k if imageand image

Proposition 5.1

Let (A, μ {.,.} ,ε ) be a color Poisson algebra and t α an even color Poisson algebra endomorphism of the form imagewhere imageare endomorphism of A (as color Poisson algebra), t is a parameter in K and k is an entiger.Let imageand imagethen imageis 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]).

Modules Over Color Hom-Poisson Algebras

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 image 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 imagecalled structure map, such that

image (82)

image(83)

image(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 imageDefine the map

image

Then image 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 image

image(89)

image(90)

image (91)

image(92)

for any image

Remark 6.1

When imagewe 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 image

image(96)

Then imageis 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 imageFor any image

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 imageWe have

image

This establishes the Theorem.

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

Corollary 6.1

Let image 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, forimage we have

image

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

image

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

Proof

We know from Lemma (6.1) that imageis an A-module structure map. And, for image imageone has

image

This is similar to the relation 21 for image

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 imageand imagesuch that

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

(ii) And for any image

image

image

When image

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 imageis a module over A with structure maps imageimageand image

Defined by

image

for any image

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

image(125)

image(126)

Define another color Hom-Poisson module structure on M.

Proof

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

image

And, for image

image

Hence equations 23 and 24 hold for imageand imageThis 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 image Then image, imagedefine another color Hom- Poisson module structure on M.

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