Representations and Deformations of Hom-Lie-Yamaguti Superalgebras

Shuangjian Guo,Xiaohui Zhang,Shengxiang Wang

doi:10.1155/2020/9876738

Abstract

Let L,α 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 L,α and present some properties. Finally, we investigate the deformations of L,α by choosing some suitable cohomology.

Highlights

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]

Summary

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”. 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 [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]. The purpose of this paper is to study the representations and deformations of Hom-Lie-Yamaguti superalgebras.

Preliminaries

Representations of Hom-LieYamaguti Superalgebras