The classical Lie-Yamaguti Yang-Baxter equation and Lie-Yamaguti bialgebras
The classical Lie-Yamaguti Yang-Baxter equation and Lie-Yamaguti bialgebras
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DOI：
10.1360/scm-2022-0517
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发表时间：
2022-10
影响因子：
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通讯作者：
Jia Zhao;Yu Qiao
中科院分区：
文献类型：
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作者：
Jia Zhao;Yu Qiao
本文发展了Lie-Yamaguti代数的双代数理论。为此，我们利用两种类型的相容性条件：局部上循环条件和双重构造。在Lie-Yamaguti代数中定义了经典的Yang-Baxter方程，并证明了经典的Yang-Baxter方程的解对应于关于余伴随表示的相对Rota-Baxter算子.进一步地，我们将Bai [1]和Semonov-Tian-Shansky [19]的一些结果推广到Lie-Yamaguti代数的情形.然后引入李-山古提双代数的匹配对的概念，并在此基础上引入李双代数的Manin三元组方法，得到了二重构造李-山古提双代数的概念.证明了Lie-Yamaguti代数的匹配对、Manin三元组和二重构造Lie-Yamaguti双代数是等价的。最后，我们阐明了局部上循环条件是Lie-Yamaguti双代数双重构造的一个特例。
In this paper, we develop the bialgebra theory for Lie-Yamaguti algebras. For this purpose, we exploit two types of compatibility conditions: local cocycle condition and 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 by Bai in [1] and Semonov-Tian-Shansky in [19] 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.