Bakayoko I* and Diallo OW
Department of Mathematics, UJNK/Nzerekore University Centre, PO Box 50, Nzerekore, Guinea
Received date: January 23, 2015 Accepted date: July 25, 2015 Published date: July 29, 2015
Citation: Bakayoko I, Diallo OW (2015) Some Generalized Hom-Algebra Structures. J Generalized Lie Theory Appl 9:226. doi:10.4172/1736-4337.1000226
Copyright: © 2015 Bakayoko I, et al. 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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In this paper, we introduce generalized left-Hom-symmetric algebras and generalized Hom-dendriform algebras as well as the corresponding modules. We investigate the connection between these categories of generalized Homalgebras and modules. We give various constructions of these generalized Hom-algebra structures from either a given one or an ordinary one. We prove that any generalized Hom-dialgebras give rise to generalized Hom-Leibniz- Poisson algebras and generalized Hom-Poisson dialgebras.
Generalized Hom-associative; Hom-lie; Homdendriform; Hom-Leibniz-Poisson algebras; Generalized Homdialgebras; Graded modules
Hom-algebraic structures were appeared for the first time in the works of Aizawa and Sato  as a generalization of Lie algebras. Other interesting Hom-type algebraic structures of many classical structures were studied as Hom-associative algebras, Hom-Lie admissible algebras and more general G-Hom-associative algebras , n-ary Hom-Nambu-Lie algebras , Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras , Hom-alternative algebras, Hom-Malcev algebras and Hom-Jordan algebras , L-modules, L-comodules and Hom-Lie quasi-bialgebras , Laplacian of Hom-Lie quasi- bialgebras . Hom-algebraic structures were extended to the case of G-graded Lie algebras by studying Hom-Lie superalgebras, Hom-Lie admissible superalgebras , color Hom-Lie algebras , color Hom-Lie bialgebras and color Hom-Poisson bialgebras  and color Hom- Poisson algebras. Color Hom-Poisson algebras were introduced  as generalization of Hom-Poisson algebras . It is shown that every color Hom-associative algebra has a non-commutative Hom-Poisson algebra structure in which the Hom-Poisson bracket is the commutator bracket. It is also proved that twisting a color Hom-Poisson module structure map by a color Hom-Poisson algebra endomorphism, we get another one. Color Hom-Poisson algebras generalize color Homassociative algebras [9,12] color Hom-Lie algebras [9,12] and Hom- Lie superalgebras . The author  presents some constructions of quadratique color Hom-Lie algebras, and this is used to provide several examples. T*-extensions and central extensions of color Hom-Lie algebras and some cohomological characterizations are established . These structures are well-known to physicists and to mathematicians studying differential geometry and homotopy theory. The cohomology theory of Lie superalgebras  has been generalized to the cohomology of Hom-Lie superalgebras .
The purpose of this paper is to introduce some generalized (color or graded) Hom-algebras and the corresponding modules. The paper is organized as follows. In section 2, we recall some basic notions related to color Hom-associative algebras and color Hom- Lie algebras. In section 3, we define generalized left-Hom-symmetric algebras (resp. generalized Hom-dendriform algebras) and prove that to any generalized left-Hom-symmetric algebra one may associate a generalized Hom-Lie algebra (resp. a generalized Hom-dendriform algebra and a generalized Hom-associative algebra). Next, we prove that any generalized left-symmetric algebra can be deformed into a generalized left-Hom-symmetric algebra along with any even linear selfmap. In section 4, we point out that generalized Hom-dialgebras give rise to generalized non-commutative Hom-Leibniz-Poisson algebras and generalized Hom-Poisson dialgebras. A construction theorem of generalized Hom-Poisson dialgebras from generalized Poisson dialgebras and an endomorphism is given. In section 5, we point out that bimodules over a generalized left-Hom-symmetric algebra S (resp. generalized Hom-dendriform algebra D) extend to modules over the generalized Hom-Lie algebra (resp. bimodules over the generalized left-Hom-symmetric algebra, generalized Hom-associative algebra) associated to S (resp. D).
Throughout this paper, all graded vector spaces are assumed to be over a field K of characteristic different from
Let G be an abelian group. A vector space V is said to be a G-graded if, there exists 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 be another G-graded algebra. A morphism of G-graded algebras is by definition an algebra morphism from A to which is, in addition an even mapping.
Definition 2.1: Let G be an abelian group. A map is called a skew- symmetric bicharacter on G if the following identities hold,
(i) (ii) (iii)
a, b, c ∈ 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 2.2: A generalized Hom-associative algebra is a quadruple (A, μ, ε, α) consisting of a G-graded vector space A, an even bilinear map μ: A × A → A, a bicharacter ε and an even linear map α: A → A such that
μ(α(x), μ(y, z)) = μ(μ(x, y), α(z)). (1)
If in addition μ(x, y) = ε(x, y)μ(y, x), the generalized Hom-associative algebra (A, μ, ε, α) is said to be a ε-commutative generalized Homassociative algebra.
Remark 2.1: When α = Id, we recover the classical generalized associative algebra.
Recall that the (generalized) Hom-associator, asA, of a Hom-algebra A is defined as
Observe that asA ≡0 when A is a generalized Hom-associative algebra.
Definition 2.3:A generalized Hom-Lie algebra is a quadruple (A, [−, −], ε, α) consisting of a G-graded vector space A, an even bilinear map [−, −] : A × A → A (i.e. [Aa, Ab] ⊆ Aa+b for all a, b ∈ G), a bicharacter ε, and an even linear map α: A → A such that for any , we have
(2) (3) (4)
Where means cyclic summation.
Remark 2.2: A generalized Lie algebra (A, [−, −], ε) is a generalized Hom-Lie algebra with α = Id.
This section is devoted to generalized left-Hom-symmetric algebras and generalized Hom- dendriform algebras which are the twisted analogue of generalized left-symmetric algebras  and Homdendriform algebras [18,19] respectively.
The following lemma connects generalized Hom-associative algebras to generalized Hom-Lie algebras.
Lemma 3.1: Let (A, μ, ε, α) be a generalized Hom-associative algebra .
Then is a generalized Hom-Lie algebra, denoted by L(A).
Definition 3.1: Let G be an abelian group and ε a bicharacter on G. A graded non-associative algebra with the even linear map α: S → S and the multiplication satisfying
is called a generalized left-Hom-symmetric algebra if the following generalized left-Hom- symmetric identity (or ε-left-Hom-symmetric identity) is satisfied
In term of the (generalized) Hom-associator, the ε-left-Homsymmetric identity may written as
Definition 3.2: Let (S, μ, ε, α) and be two generalized Hom-left-symmetric algebras. An even linear map is said to be a morphism of generalized Hom- left-symmetric algebras if and
for all x, y ∈ S.
Theorem 3.1: Let be a generalized left-Hom-symmetric algebra and a left-Hom-symmetric algebra endomorphism. Then, for any non-negative entiger n, is a generalized left-Hom-symmetric algebra, where
Moreover, suppose that is another generalized leftsymmetric algebra and a left-Hom-symmetric algebras endorphism. is a generalized left-symmetric algebras morphism that satisfies , then
is a morphism of generalized left-Hom-symmetric algebras.
Proof: For any
For the second part, we have
This completes the proof.
Lemma 3.2: Let (S, â—¦, ε, α) be a generalized left-Hom-symmetric algebra. Then, for any the commutator
makes S to become a generalized Hom-Lie algebra, denoted L(S ). Proof: For any we have
Using the ε-left-Hom-symmetric identity, observe that the following sums are This shows that
Definition 3.3: A generalized Hom-dendriform algebra is a G-graded vector space D together with two even bilinear maps and an even linear map α : S → S such that
(9) (10) (11)
If is another generalized Hom-dendriform algebra and is an even linear map such that then f is said to be a morphism of generalized Hom-dendriform algebras when it is compatible with both products and
When α = Id, we obtain morphism of generalized dendriform algebras.
In a generalized Hom-dendriform algebra if the twisting map α is morphism, then is said to be a multiplicative generalized Hom-dendriform algebra.
The below Theorem produces generalized Hom-dendriform algebras from generalized dendriform algebras.
Theorem 3.2: Let be a generalized dendriform algebra and α: D → D an even linear map such that
Then, for any non-negative entiger n is a generalized multiplicative Hom-dendriform algebra.
Suppose that is another generalized dendriform algebra and an even linear map that satisfies and If is a morphism of generalized dendriform algebras, then morphism of generalized Hom-dendriform algebra.
Proof: The three axioms are easy to check. For example, for any
This proves (9). The other two axioms are proved similarly. The second part is showed likewise the proof of the second part of Theorem 3.1.
Lemma 3.3: Let be a generalized Hom-dendriform algebra. Then, D is a generalized left-Hom-symmetric algebra with the structure map
x, y∈D .
Proof: For any , we have
The term a1 + a2 + a3 vanish (11), b1 + b2 + b3 vanish (11), c1 + c2 + c3 vanishes (10), d1 + d2 + d3 vanishes (10), e1 + e2 vanishes (9), f1 + f2 vanishes (9). Thus, is a generalized Hom-left-symmetric algebra.
Lemma 3.4: Let be a generalized Hom-dendriform algebra. Then, is a generalized Hom-associative algebra with the multiplication given by
Proof: For any
The left hand side vanishes by axioms in Definition 3.3. Therefore, the conclusion holds.
Let us denote by GHDA: the category of generalized Homdendriform algebras, GLHSA: the category of generalized left-Homsymmetric algebras, GHAA: the category of generalized Homassociative algebras and
GHLA: the category of generalized Hom-Lie algebras. Then, the above discussion may be summarized in the following theorem.
Theorem 3.3: The following diagram is commutative
Proof: The top horizontal arrow and the bottom horizontal arrow follow from Lemma 3.3 and Lemma 3.1 respectively. The left vertical arrow is established in Lemma 3.4 and right vertical arrow is the functor constructed in Lemma 3.2.
In this section we will show that generalized Hom-dialgebras give rise to generalized non- commutative Hom-Leibniz-Poisson algebras and generalized Hom-Poisson dialgebras.
Definition 4.1: A generalized Hom-dialgebra is a quintuple where D is a G-graded vector space, are even bilinear maps, is a bicharacter and α: D → D is an even linear map such that the following five axioms are satisfied for
Definition 4.2: A generalized non-commutative Hom-Leibniz- Poisson algebra is a G-graded vector space P together with two even bilinear maps and a bicharacter and an an even linear map α : P → P such that
i) is a generalized Hom-Leibniz algebra i.e.
ii) is a generalized Hom-associative algebra, iii) and the following identity
holds, for all
If in addition, α is an endomorphism with respect to • and [−, −], we say that is a multiplicative generalized noncommutative Hom-Leibniz-Poisson algebra.
Remark 4.1: When the Hom-associative product • is ε-commutative, then is said to be a commutative generalized Hom- Leibniz Poisson algebra.
The following result gives a connection generalized Hom-dialgebras and generalized Hom-Leibniz-Poisson algebras.
Theorem 4.1: Let be a generalized Hom-dialgebra. Define the even bilinear map by setting
Then is a generalized Hom-Leibniz-Poisson algebra.
Proof: We write down all the twelve terms involved in the Hom- Leibniz identity :
Using the five axioms in Definition 4.1, it is immediate to see that (12) hols. And,
The left hand side vanishes by axioms in Definition 4.1.
In the sequel, we define generalized Hom-Poisson dialgebras, and we give a construction theorem.
Definition 4.3: A generalized Hom-Poisson dialgebra is a sextuple in which P is a G-graded vector space are three even bilinear maps, is a bicharacter and α: P → P is an even linear map such that
(14) (15) (16)
Example 4.1: Any Hom-Poisson dialgebra  is a generalized Hom-Poisson dialgebra with trivial grading.
Theorem 4.2: Let be a generalized Hom-dialgebra. Then is a generalized Hom-Poisson dialgebra, where
Proof: According to the second part of the proof of Theorem 4.1, we only need to prove (15) and (16). For any we have
This completes the proof.
Definition 4.4: Let and be two Hom-Poisson dialgebras. An even linear map is said to a morphism of generalized Hom- Poisson dialgebras, if and for any
The following theorem allows obtaining a generalized Hom-Poisson dialgebra from generalized Poisson dialgebra and an endomorphism.
Proposition 4.1: Let be a generalized Poisson dialgebra and α: D → D an endomorphism of generalized Poisson dialgebras. Then, is a generalized Hom-Poisson dialgebra, with
Proof: The proof is analogue to the one of Theorem 3.2.
In this section, we introduce modules over some generalized Homalgebras and study the connection between them.
Recall the following definitions.
Definition 5.1: Let G be an abelian group. A Hom-module is a pair (M, β) in which M is a G-graded vector space and is an even linear map .
Definition 5.2: Let (A, μ, ε, α) be a generalized Hom-associative algebra. An A-module is a Hom-module (M, β) together with an even bilinear map called structure map , such that
(17) (18) (19)
Definition 5.3: Let be a generalized Hom-Lie algebra and (M, β) a Hom-module. The G-graded vector space M is said to be a module over L, if it is endowed with an even bilinear map  such that
The following lemma shows that A-modules extend to L(A)- modules.
Lemma 5.1: Let (A, μ, ε, α) be a generalized Hom-associative algebra and (M, β) an A-module with structure map . Then, M is an L(A)-module with the same structure map .
Definition 5.4: Let be a generalized left-Homsymmetric algebra. An S–bimodule is a G-graded vector space M endowed with an even linear map β: M → M, two even bilinear maps and such that
Proposition 5.1: Let be a generalized left-Homsymmetric algebra and a bimodule over S. Then, M is a module over L(S) with the structure map defined by
Proof: For any we have
Using (21) observe that a1 + a2 + a3 + a4 = 0, and b1 + b2 + b3 + b4 = 0 and c1 + c2 + c3 + c4 = 0 by (22). The two other relations of the Definition 5.3, are straight forward proved. This prove that M is a module over L(S).
Corollary 5.1: Let be a left-Hom-symmetric algebra and a bimodule over S. Then, M is a module over L(S) with the structure map defined by
Definition 5.5: Let be a generalized Hom-dendriform algebra. A D-bimodule is a graded vector space M together with an even linear map β: M → M and four even bilinear maps
such that (28) (29) (30) (31) (32) (33) (34) (35) (36)
Proposition 5.2: Let be a bimodule over the generalized Hom-dendriform algebra Then, M is a bimodule over the generalized left-Hom-symmetric algebra,
associated to D, S ym(D), with the structure map defined by
Proof: We only need to prove (21) and (22). The other relations being obvious. So, for any , we have
Observe that a1 + a2 + a3 cancels by (30), b1 + b2cancels by (31),c1 + c2cancels by (31), d1 + d2 + d3 cancels by (35), e1 + e2 + e3 cancels by (30) and f1 + f2 + f3 cancels by (35).
Observe that a1 + a2 + a3 = 0 by (33), b1 + b2 = 0 by (28), c1 + c2 = 0 by (34), d1 + d2 + d3 = 0 by (32), e1 + e2 + e3 = 0 by (36) and f1 + f2 + f3 = 0 by (29).
Corollary 5.2: Let be a bimodule over the Homdendriform Then, M is a bimodule over the left-Homsymmetric algebra Sym(D) associated to D.
Definition 5.6: Let be a Hom-associative algebra and let (M, β) be a Hom-module. A bimodule structure on M consists of :
i) a left A-action and
ii) a right A-action such that the following conditions hold for x, y ∈ A and m ∈ M
(38) (39) (40) (41) (42)
Proposition 5.3: Let be a bimodule over the Homdendriform Then, M is a bimodule over the Hom-associative algebra associated to D, Ass(D), with the structure map
Proof: For any we have
The left hand side vanishes by axioms (28)-(30), with trivial grading.
The left hand side vanishes by axioms (34)-(36), with trivial grading. Finally,
The left hand side vanishes by axioms (31)-(33), with trivial grading.
Now, let us denote by HDbiM: the category of bimodules over Hom-dendriform algebra, LHSbiM: the category of bimodules over left-Hom-symmetric algebra, HABiM: the category of bimodules over Hom-associative algebra and HLM: the category of modules over Hom-Lie algebra. Then, we have:
Theorem 5.1: The following diagram of categories and functors is commutative
Proof: The top horizontal arrow and the bottom horizontal arrow follow from Corollary 5.2 and Lemma 5.1 respectively. The left vertical arrow is established in Proposition 5.3 and right vertical arrow is the functor constructed in Corollary 5.1.
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