从Leibniz超代数到Lie-Yamaguti超代数

FROM LEIBNIZ SUPERALGEBRAS TO LIE-YAMAGUTI

SUPERALGEBRAS

TANG Xin-xin1,ZHANG Qing-cheng1,WANG Chun-yue2

【摘 要】Abstract:In this paper,we study the construction of Lie-Yamaguti superalgebras.By using left Leibniz superalgebras,we give the construction of left Leibniz superalgbebras,then give the construction of Lie-Yamaguti superalgebras from left Leibniz superalgebras.So we gain the construction of Lie-Yamaguti superalgebras,which generalizes the construction of Lie-Yamaguti algebras in the situation of superalgebras. 【期刊名称】数学杂志 【年(卷),期】2018(038)004 【总页数】13

【关键词】Keywords: Lie-Yamaguiti superlagebras;(left)Leibniz superlagebras;Akivis superalgebras;Lie supertriple systems;construction 2010 MR Subject Classification:17A30;17A32;17D99 Document code:A

Article ID:0255-7797(2018)04-0589-13 ?Received date:2017-03-03 Accepteddate:2017-04-26

Foundation item:Supported by the NSFC(11471090).

1 Introduction

Lie algebras were studied for many years in mathematics and

physics,such as in quantum field theory.As the noncommutative analogs of Lie algebras,Leibniz algebras were first introduced by Cuvier and Loday in[1]and[2].Researchers obtained many results about Leibniz algebras and we can find some of them in[3–6].There are two kinds of Leibniz algebras,left Leibniz algebras and right Leibniz algebras[7].For a given

non-commutative

algebra(A,·),if

the

left

multiplication

lx ·y=x ·y,?x,y ∈ A is a derivation of A,then(A,·)is called a left Leibniz algebra[8].As non-associative algebras,left Leibniz algebras can construct Akivis algebras[9].Kinyon and Weinstein found that a left Leibniz algebra has a Lie-Yamaguti algebra structure by using an enveloping Lie algebra of Leibniz algebras.

Recently,Leibniz algebras are generalized to Leibniz superalgebras by Dzhumadil in[10].Then some important results were obtained such as[11]and[12].Like left Leibniz algebras and right Leibniz algebras,we can similarly obtain left Leibniz superalgebras and right Leibniz superalgebras.If(A,·)is a left Leibniz superalgebra,we can obtain a right Leibniz superalgebra(A,?)by defining x ? y=(?1)|x||y|y ·x.In this paper,we study the construction of left Leibniz superalgebras and Lie-Yamaguti superalgebras.

This paper is organized as follows.In Section 2,we recall the definition of Leibniz

superalgebras

and

prove

that

every

non-associative

superalgebra has an Akivis superalgebra structure.Then we give

examples and constructions of left Leibniz superalgebras.In Section 3,we define Lie-Yamaguti superalgebras and prove that every left Leibniz superalgebra has a Lie-Yamaguti superalgebra structure.

Throughout this paper,K denotes a field of characteristic zero;All vector spaces and algebras are over K;hg(A)denotes the set of homogeneous elements of the superalgebra A.

2 Leibniz Superalgebras

In this section,we introduce the definition of Leibniz superslgebras,and then give the constructions and examples of Leibniz superalgebras. Definition 2.1[11](i)A(left)Leibniz superalgebra is a pair(A,·),in which A is a superspace,·:A×A→A an even bilinear map such that for all x,y,z∈hg(A).

(ii)A(right)Leibniz superalgebra is a pair(A,·),in which A is a superspace,·:A×A →A an even bilinear map such that for all x,y,z∈hg(A).

In this paper, “Leibniz superalgebras” means “left Leibniz superalgebras”.A super skew-symmetric Leibniz superalgebra is a Lie superalgebra.In this case,equations(2.1)and(2.2)become the Jacobi super-identity.If(A,·)is a left Leibniz superalgebra,we can obtain a right Leibniz superalgebra(A,?)by defining x ? y=(?1)|x||y|y ·x. Definition 2.2 Let(A,·)be a superalgebra.

(i)The super-associator of A is an even trilinear map:A×A×A→A defined