Rota-Baxter operators on braces, post-braces and the Yang-Baxter equation

TL;DR

This paper introduces and explores the connections between Rota-Baxter operators, braces, post-braces, and solutions to the Yang-Baxter equation, establishing new theoretical links and a factorization theorem.

Contribution

It defines relative Rota-Baxter operators on braces and post-braces, showing their equivalence and connection to Yang-Baxter solutions, and proves a factorization theorem for enhanced operators.

Findings

Relative Rota-Baxter operators induce post-braces.

Post-braces produce solutions to the Yang-Baxter equation.

A factorization theorem for enhanced Rota-Baxter operators is established.

Abstract

Combining the notions of braces and relative Rota-Baxter operators on groups in connection with the Yang-Baxter equation and a factorization theorem of Lie groups from integrable systems, relative Rota-Baxter operators on braces and post-braces are introduced. A relative Rota-Baxter operator on a brace naturally induces a post-brace, and conversely, every post-brace determines a relative Rota-Baxter operator on its sub-adjacent brace. Furthermore, a post-brace yields two Drinfel'd-isomorphic solutions to the Yang-Baxter equation. As a special case, {\it enhanced} relative Rota-Baxter operators give rise to matched pairs of braces. Focusing on enhanced Rota-Baxter operators on two-sided braces, a corresponding factorization theorem is established. Examples are provided from the two-sided brace associated with the three-dimensional Heisenberg Lie algebra.