Representations and deformations of Hom-Lie-Yamaguti superalgebras

Let $(L, \alpha)$ be a Hom-Lie-Yamaguti superalgebra. We first introduce the representation and cohomology theory of Hom-Lie-Yamaguti superalgebras. Furthermore, we introduce the notions of generalized derivations and representations of $(L, \alpha)$ and present some properties. Finally, we investigate the deformations of $(L, \alpha)$ by choosing some suitable cohomology.


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 quasideformations of Lie algebras of vector fields, in particular qdeformations 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][7][8][9][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.
A homomorphism between two Hom-LY superalgebras ðL, αÞ and ðL′, α′Þ is a linear map φ : Example 5. Consider the 5-dimensional ℤ 2 -graded vector space L = L 0 ⊕ L 1 , over an arbitrary base filed K of characteristic different from 2, with basis fu 1 , u 2 , u 3 g of L 0 and fe 1 , e 2 g of L 1 , and the nonzero products on these elements are induced by the following relations: Define the superspace homomorphisms α : L ⟶ L by It is not hard to check that ðL, * ,αÞ is a Hom-Leibniz superalgebra. By [18], we can define ½·, · and f·, · , · g, and the nonzero products on these elements are induced by the following relations: 2 Advances in Mathematical Physics Then, ðL, ½·, · , f·, · , · g, αÞ becomes a Hom-LY superalgebra.

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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 L ⊕ V is a Hom-LY superalgebra.
Let V be a representation of Hom-LY superalgebra L. Let us define the cohomology group of L with coefficients in V. Let f : L × L × ⋯ × L ⟶ V be an n-linear map such that the following conditions are satisfied: The vector space spanned by such linear maps is called an n-cochain of L, which is denoted by C n ðL, VÞ for n ≥ 1.
Definition 10. For the case n ≥ 2, the ð2n, 2n + 1Þ-cohomology group of a Hom-LY superalgebra L with coefficients in V 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 L with coefficients in V.

α k -Derivations of Hom-Lie-Yamaguti Superalgebras
In this section, we give the definition of α k -derivations of Hom-LY superalgebras, then, we study its generalized derivations.

Definition 11. A linear map
for all x, y, z ∈ L, where |D | denotes the degree of D.
We denote by DerðLÞ = ⊕ k≥0 Der α k ðLÞ, where Der α k ðLÞ is the set of all homogeneous α k -derivations of L. Obviously, DerðLÞ is a subalgebra of EndðLÞ.

Theorem 12.
DerðLÞ is a Lie superalgebra, where the bracket product is defined as follows: Proof. It is sufficient to prove ½Der α k ðLÞ, Der α s ðLÞ ⊆ Der α k+s ðLÞ. Note that It follows that ½D, D ′ ∈ Der α k+s ðLÞ.
Definition 13. Let ðL, αÞ be a Hom-LY superalgebra. D ∈ En d s ðLÞ is said to be a homogeneous generalized for all x, y, z ∈ L.
Definition 14. Let ðL, αÞ be a Hom-LY superalgebra. D ∈ En d s ðLÞ is said to be a homogeneous α k -quasiderivation of L, if there exist endomorphisms D′, D′ ′ ∈ End s ðLÞ such that for all x, y, z ∈ L.
Let GDerðLÞ and QDerðLÞ be the sets of homogeneous generalized α k -derivations and of homogeneous α k -quasiderivations, respectively. That is, Definition 15. Let ðL, αÞ be a Hom-LY superalgebra. The α k -centroid of L is the space of linear transformations on L given by We denote CðLÞ = ⨁ k≥0 C α k ðLÞ and call it the centroid of L.
Definition 16. Let ðL, αÞ be a Hom-LY superalgebra. The quasicentroid of L is the space of linear transformations on L given by for all x, y, z ∈ L. We denote QCðLÞ = ⊕ k≥0 QC α k ðLÞ and call it the quasicentroid of L.
Remark 17. Let ðL, αÞ be a Hom-LY superalgebra. Then C ðLÞ ⊆ QCðLÞ: Definition 18. Let ðL, αÞ be a Hom-LY superalgebra. D ∈ EndðLÞ is said to be a central for all x, y, z ∈ L. Denote the set of all central α k -derivations by ZDerðLÞ.

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(2) ZDerðLÞ is an ideal of DerðLÞ. Proof.
(2) For any D 1 ∈ ZDer α k ðLÞ, D 2 ∈ Der α s ðLÞ and x, y, z ∈ L, we have 7 Advances in Mathematical Physics Also, we have and it is easy to check that It follows that ½D 1 , D 2 ∈ ZDer α k+s ðLÞ. That is, ZDerðLÞ is an ideal of DerðLÞ.