Abstract

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.

1. Introduction

Lie triple systems arose initially in Cartan’s study of Riemannian geometry. Jacobson [1] 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 [2] to give an algebraic interpretation of the characteristic properties of the torsion and curvature of homogeneous spaces with canonical connection in [3]. 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 [4].

The theory of Hom-algebra started from Hom-Lie algebras introduced and discussed in [5], 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

So far, many authors have studied Hom-type algebras motivated in part for their applications in physics ([6–10]).

In [11], 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 [12], Ma et al. studied the formal deformations of Hom-Lie-Yamaguti algebras. Recently, in [13], Lin et al. introduced the quasi-derivations of Lie-Yamaguti algebras. In [14], 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 [15]. In [16], Zhang et al. introduced the notion of crossed modules for Hom-Lie-Yamaguti algebras and studied their construction of Hom-Lie-Yamaguti algebras.

In [17], Gaparayi et al. introduced the concept of Hom-Lie-Yamaguti superalgebras and gave some examples of Hom-Lie-Yamaguti superalgebras. Later, in [18], 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.

2. Preliminaries

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 [17].

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
(SHLY1)
(SHLY2)
(SHLY3)
(SHLY4)
(SHLY5)

(SHLY7)
(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 [18], 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:
(SHR1)
(SHR2)
(SHR3)
(SHR4)
(SHR5)
(SHR6)
(SHR7)
(SHR8)
(SHR9)
(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 [14].

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 Similarly, we can check that It follows that .

Definition 13. Let be a Hom-LY superalgebra. is said to be a homogeneous generalized -derivation of , if there exist three endomorphisms such that for all .

Definition 14. Let be a Hom-LY superalgebra. is said to be a homogeneous -quasiderivation of , if there exist endomorphisms such that for all .

Let and be the sets of homogeneous generalized -derivations and of homogeneous -quasiderivations, respectively. That is,

Definition 15. Let be a Hom-LY superalgebra. The -centroid of is the space of linear transformations on given by We denote and call it the centroid of .

Definition 16. Let be a Hom-LY superalgebra. The quasicentroid of is the space of linear transformations on given by for all . We denote and call it the quasicentroid of .

Remark 17. Let be a Hom-LY superalgebra. Then

Definition 18. Let be a Hom-LY superalgebra. is said to be a central -derivation of if for all . Denote the set of all central -derivations by .

Remark 19. Let be a Hom-LY superalgebra. Then

Definition 20. Let be a Hom-LY superalgebra. If , then is called the center of .

Proposition 21. Let be a Hom-LY superalgebra, then the following statements hold: (1), and are subalgebras of (2) is an ideal of .

Proof. (1)We only prove that is a subalgebra of , and similarly for cases of and . For any and , we have Similarly, we have It follows that and it is easy to check that Obviously, and are contained in , thus , that is, is a subalgebra of . (2)For any and , we haveAlso, we have and it is easy to check that It follows that . That is, is an ideal of .

Lemma 22. Let be a Hom-LY superalgebra, then the following statements hold:

Proof. (1)–(4) are easy to prove and we omit them, we only check (5). In fact, let . Then, there exist , for any , we have Thus, for any , we have Therefore, .

Proposition 23. Let be a Hom-LY superalgebra, then is a subalgebra of .

Proof. By Lemma 22, (3) and (5), we have and it follows that It is easy to verify that by the Jacobi identity of Hom-Lie algebras. Thus, The proof is finished.

Theorem 24. Let be a Hom-Lie-Yamaguti superalgebra, where is surjective, then . Moreover, if , then .

Proof. For any and , since is surjective, there exist such that , then we have So and therefore, . Moreover, if , it is easy to see that .

5. 1-Parameter Formal Deformations of Hom-Lie-Yamaguti Superalgebras

Let be a Hom-LY superalgebra over and the power series ring in one variable with coefficients in . Assume that is the set of formal series whose coefficients are elements of the vector space .

Definition 25. Let be a Hom-LY superalgebra. A 1-parameter formal deformations of is a pair of formal power series of the form where each is a -bilinear map (extended to be -bilinear) and each is a -trilinear map (extended to be -trilinear) such that is a Hom-Lie-Yamaguti superalgebra over . Set and , then and can be written as , respectively.
Since is a Hom-LY superalgebra. Then, it satisfies the following axioms: for all .

Remark 26. Equations (42)–(49) are equivalent to () for all . These equations are called the deformation equations of a Hom-LY superalgebra. Equations (50)–(53) imply .

Let in Equations (50)–(57). Then, which imply , i.e.,

The pair is called the infinitesimal deformation of .

Definition 27. Let be a Hom-LY superalgebra. Two 1-parameter formal deformations and of are said to be equivalent, denoted by , if there exists a formal isomorphism of -modules where is a -linear map (extended to be -linear) such that

In particular, if then is called the null deformation. If , then is called the trivial deformation. If every 1-parameter formal deformation is trivial, then is called an analytically rigid Hom-LY superalgebra.

Theorem 28. Let and be the two equivalent 1-parameter formal deformations of . Then, the infinitesimal deformations and belong to the same cohomology class in

Proof. By the assumption that and are equivalent, there exists a formal isomorphism of -modules satisfying for any Comparing the coefficients of in both sides in each of the equations above, we have It follows that as desired. The proof is completed.

Theorem 29. Let be a Hom-LY superalgebra with , then is analytically rigid.

Proof. Let be a 1-parameter formal deformation of . Suppose , . Set in Equations (50)–(53), it follows that Then, there exists such that .
Consider then is a linear isomorphism and . Thus, we can define another 1-parameter formal deformation by in the form of Set and use the fact , then we have that is By the above equation, it follows that Therefore, we deduce So . Similarly, we have . By induction, we have , that is, is analytically rigid. The proof is finished.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The paper is supported by the NSF of China (No. 11761017), the Guizhou Provincial Science and Technology Foundation (No. [2020]1Y005), and the Anhui Provincial Natural Science Foundation (Nos. 1908085MA03 and 1808085MA14).