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

TL;DR

This paper develops the theory of Lie-Yamaguti bialgebras, introducing the classical Yang-Baxter equation, and establishing equivalences between matched pairs, Manin triples, and double constructions within this algebraic framework.

Contribution

It extends bialgebra theory to Lie-Yamaguti algebras, defining the classical Yang-Baxter equation and establishing key structural equivalences.

Findings

Classical Yang-Baxter equation solutions correspond to relative Rota-Baxter operators.

Matched pairs, Manin triples, and double constructions are equivalent in Lie-Yamaguti algebras.

Local cocycle condition is a special case of double construction.

Abstract

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.