Rota-Baxter groups, skew left braces, and the Yang-Baxter equation

TL;DR

This paper explores the relationship between Rota-Baxter groups and skew left braces, showing how they are interconnected and how each can be embedded into the other, advancing the understanding of algebraic structures related to the Yang-Baxter equation.

Contribution

It establishes a connection between Rota-Baxter groups and skew left braces, demonstrating embeddings and conditions under which they are equivalent or related.

Findings

Every Rota-Baxter group induces a skew left brace.

Every skew left brace can be embedded into a Rota-Baxter group.

Complete additive groups of skew left braces correspond to Rota-Baxter groups.

Abstract

Braces were introduced by W. Rump in 2006 as an algebraic system related to the quantum Yang-Baxter equation. In 2017, L. Guarnieri and L. Vendramin defined for the same purposes a more general notion of a skew left brace. Recently, L. Guo, H. Lang, Y. Sheng [arXiv:2009.03492] gave a definition of what is a Rota-Baxter operator on a group. We connect these two notions as follows. It is shown that every Rota-Baxter group gives rise to a skew left brace. Moreover, every skew left brace can be injectively embedded into a Rota-Baxter group. When the additive group of a skew left brace is complete, then this brace is induced by a Rota-Baxter group. We interpret some notions of the theory of skew left braces in terms of Rota-Baxter operators.