Yang-Baxter equations and relative Rota-Baxter operators for left-Alia algebras associated to invariant theory

TL;DR

This paper explores the structure of left-Alia algebras, introduces the left-Alia Yang-Baxter equation, and establishes connections between solutions of this equation and the construction of left-Alia bialgebras, advancing the algebraic theory related to invariant theory.

Contribution

It introduces the left-Alia Yang-Baxter equation and links its antisymmetric solutions to the construction of triangular left-Alia bialgebras, along with new notions of Rota-Baxter operators.

Findings

Antisymmetric solutions of the left-Alia Yang-Baxter equation produce triangular left-Alia bialgebras.

The paper defines relative Rota-Baxter operators for left-Alia algebras.

Connections between invariant theory and the structure of left-Alia algebras are established.

Abstract

Left-Alia algebras are a class of algebras with symmetric Jacobi identities. They contain several typical types of algebras as subclasses, and are closely related to the invariant theory. In this paper, we study the construction theory of left-Alia bialgebras. We introduce the notion of the left-Alia Yang-Baxter equation. We show that an antisymmetric solution of the left-Alia Yang-Baxter equation gives rise to a left-Alia bialgebra that we call triangular. The notions of relative Rota-Baxter operators of left-Alia algebras and pre-left-Alia algebras are introduced to provide antisymmetric solutions of the left-Alia Yang-Baxter equation.