Research Article | Open Access
Shuangjian Guo, Xiaohui Zhang, Shengxiang Wang, "Representations and Deformations of Hom-Lie-Yamaguti Superalgebras", Advances in Mathematical Physics, vol. 2020, Article ID 9876738, 12 pages, 2020. https://doi.org/10.1155/2020/9876738
Representations and Deformations of Hom-Lie-Yamaguti Superalgebras
Let be a Hom-Lie-Yamaguti superalgebra. We first introduce the representation and cohomology theory of Hom-Lie-Yamaguti superalgebras. Also, we introduce the notions of generalized derivations and representations of and present some properties. Finally, we investigate the deformations of by choosing some suitable cohomology.
Lie triple systems arose initially in Cartan’s study of Riemannian geometry. Jacobson  first introduced Lie triple systems and Jordan triple systems in connection with problems from Jordan theory and quantum mechanics, viewing Lie triple systems as subspaces of Lie algebras that are closed relative to the ternary product. Lie-Yamaguti algebras were introduced by Yamaguti in  to give an algebraic interpretation of the characteristic properties of the torsion and curvature of homogeneous spaces with canonical connection in . He called them generalized Lie triple systems at first, which were later called “Lie triple algebras”. Recently, they were renamed as “Lie-Yamaguti algebras” in .
The theory of Hom-algebra started from Hom-Lie algebras introduced and discussed in , motivated by quasi-deformations of Lie algebras of vector fields, in particular -deformations of Witt and Virasoro algebras. More precisely, Hom-Lie algebras are different from Lie algebras as the Jacobi identity is replaced by a twisted form using morphism. This twisted Jacobi identity is called Hom-Jacobi identity given by
In , Gaparayi and Issa introduced the concept of Hom-Lie-Yamaguti algebras, which can be viewed as a Hom-type generalization of Lie-Yamaguti algebras. In , Ma et al. studied the formal deformations of Hom-Lie-Yamaguti algebras. Recently, in , Lin et al. introduced the quasi-derivations of Lie-Yamaguti algebras. In , Zhang and Li introduced the representation and cohomology theory of Hom-Lie-Yamaguti algebras and studied deformations and extensions of Hom-Lie-Yamaguti algebras as an application, generalizing the results of . In , Zhang et al. introduced the notion of crossed modules for Hom-Lie-Yamaguti algebras and studied their construction of Hom-Lie-Yamaguti algebras.
In , Gaparayi et al. introduced the concept of Hom-Lie-Yamaguti superalgebras and gave some examples of Hom-Lie-Yamaguti superalgebras. Later, in , Gaparayi et al. studied the relation between Hom-Leibniz superalgebras and Hom-Lie-Yamaguti superalgebras.
The purpose of this paper is to study the representations and deformations of Hom-Lie-Yamaguti superalgebras. This paper is organized as follows. In Section 2, we recall the definitions of Hom-Lie-superalgebras and Hom-Lie supertriple systems. In Section 3, we introduce the representation and cohomology theory of Hom-Lie-Yamaguti superalgebras. In Section 4, we introduce the notions of generalized derivations and representations of a Hom-Lie-Yamaguti superalgebra and present some properties. In Section 5, we consider the theory of deformations of a Hom-Lie-Yamaguti superalgebra by choosing a suitable cohomology.
Throughout this paper, we work on an algebraically closed field of characteristic different from 2 and 3, all elements like should be homogeneous unless otherwise state. We recall the definitions of Hom-Lie-superalgebras and Hom-Lie supertriple systems from [6, 9].
Definition 1. A Hom-Lie superalgebra is a -graded algebra , endowed with an even bilinear mapping and a homomorphism satisfying the following conditions, for all and :
Definition 2. A Hom-Lie supertriple system is a -graded vector space , endowed with an even trilinear mapping and a homomorphism satisfying the following conditions, (1)(2)(3)(4)for all , and denotes the degree of the element .
3. Representations of Hom-Lie-Yamaguti Superalgebras
We recall the basic definition of Hom-Lie-Yamaguti superalgebras from .
Definition 3. A Hom-Lie-Yamaguti superalgebra (Hom-LY superalgebra for short) is a quadruple in which is -vector superspace, a binary superoperation and a ternary superoperation on , and an even linear map such that
(SHLY8) for all and where denotes the sum over cyclic permutation of , and denotes the degree of the element . We denote a Hom-LY superalgebra by .
Remark 4. (1)If , then the Hom-LY superalgebra reduces to a LY superalgebra (see (SLY 1)-(SLY 6)).(2)If , for all , then becomes a Hom-Lie supertriple system .(3)If for all , then the Hom-LY superalgebra becomes a Hom-Lie superalgebra .A homomorphism between two Hom-LY superalgebras and is a linear map satisfying and
Example 5. Consider the 5-dimensional -graded vector space , over an arbitrary base filed of characteristic different from 2, with basis of and of , and the nonzero products on these elements are induced by the following relations:
Define the superspace homomorphisms by
It is not hard to check that is a Hom-Leibniz superalgebra. By , we can define and , and the nonzero products on these elements are induced by the following relations:
Then, becomes a Hom-LY superalgebra.
Definition 6. Let be a Hom-LY superalgebra and be a Hom-vector super-space. A representation of on consists of an even linear map End() and even bilinear maps End() such that the following conditions are satisfied:
(SHR10) for any . In this case, is also called an -module.
Proposition 7. Let be a Hom-LY superalgebra and be a Hom-vector superspace. Assume we have a map from to End() and maps End() satisfying (SHR1)–(SHR10). Then, is a representation of on if and only if is a Hom-LY superalgebra under the following maps: for any and .
Proof. It is easy to check that the conditions (SHLY1)–(SHLY4) hold, we only verify that conditions (SHLY5)–(SHLY8) hold for maps defined on .
For (SHLY5), we have and Thus by (SHR4), the condition (SHLY5) holds.
For (SHLY6), we have Thus by (SHR5), the condition (SHLY6) holds.
For (SHLY7), we have Thus by (SHR6), the condition (SHLY7) holds.
Now it suffices to verify (SHLY8). By the definition of the Hom-LY superalgebra, we have Thus by (SHR7), the condition (SHLY8) holds. Therefore, we obtain that is a Hom-LY superalgebra.
Let be a representation of Hom-LY superalgebra . Let us define the cohomology group of with coefficients in . Let be an -linear map such that the following conditions are satisfied:
The vector space spanned by such linear maps is called an -cochain of , which is denoted by for .
Definition 8. For any , the coboundary operator is a mapping from into defined as follows:
Proposition 9. The coboundary operator defined above satisfies , that is and .
Proof. Similar to .
The subspace of spanned by all the 's such that is called the space of cocycles while the space is called the space of coboundaries.
Definition 10. For the case , the -cohomology group of a Hom-LY superalgebra with coefficients in is defined to be the quotient space:
In conclusion, we obtain a cochain complex whose cohomology group is called the cohomology group of a Hom-LY superalgebra with coefficients in .
4. -Derivations of Hom-Lie-Yamaguti Superalgebras
In this section, we give the definition of -derivations of Hom-LY superalgebras, then, we study its generalized derivations.
Definition 11. A linear map is called an -derivation of if it satisfies for all , where denotes the degree of .
We denote by , where is the set of all homogeneous -derivations of . Obviously, is a subalgebra of .
Theorem 12. is a Lie superalgebra, where the bracket product is defined as follows:
Proof. It is sufficient to prove . Note that