经典Lie-Yamaguti Yang-Baxter方程和Lie-Yamaguti双代数  

The classical Lie-Yamaguti Yang-Baxter equation and Lie-Yamaguti bialgebras

作　　者：赵嘉 乔雨 Jia Zhao;Yu Qiao

机构地区：[1]南通大学理学院,南通226019 [2]陕西师范大学数学与统计学院,西安710119

出　　处：《中国科学：数学》2023年第10期1303-1324,共22页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11971282)资助项目。

摘　　要：本文引入Lie-Yamaguti双代数理论,给出两种类型的相容性条件:局部闭链条件和double构造.本文定义Lie-Yamaguti代数中的经典Yang-Baxter方程,并证明经典Yang-Baxter方程的解对应于关于余伴随表示的相对Rota-Baxter算子.此外,本文将Lie代数上的一些结论推广至Lie-Yamaguti代数上.然后,本文引入Lie-Yamaguti代数匹配对的概念,继而定义double Lie-Yamaguti双代数,并且证明Lie-Yamaguti代数的匹配对、Manin三元组和double Lie-Yamaguti双代数之间的等价关系.最后,本文说明局部闭链条件是double构造的一个特例.In this paper,we develop the bialgebra theory for Lie-Yamaguti algebras.For this purpose,we exploit two types of compatibility conditions:the local cocycle condition and the double construction.We define the classical Yang-Baxter equation in Lie-Yamaguti algebras and show that a solution to the classical Yang-Baxter equation corresponds to a relative Rota-Baxter operator with respect to the coadjoint representation.Furthermore,we generalize some results on Lie algebras and on 3-Lie algebras to the context of Lie-Yamaguti algebras.Then we introduce the notion of matched pairs of Lie-Yamaguti algebras,which leads us to the concept of double construction Lie-Yamaguti bialgebras following the Manin triple approach to Lie bialgebras.We prove that matched pairs,Manin triples of Lie-Yamaguti algebras,and double construction Lie-Yamaguti bialgebras are equivalent.Finally,we clarify that a local cocycle condition is a special case of a double construction for Lie-Yamaguti bialgebras.

关 键 词：Lie-Yamaguti双代数 经典Lie-Yamaguti Yang-Baxter方程 相对Rota-Baxter算子 Manin三元组 匹配对 

分 类 号：O152.5[理学—数学]