Soluble skew left braces and soluble solutions of the Yang-Baxter equation

TL;DR

This paper explores the algebraic structure of skew left braces, introducing and analyzing the concept of solubility, and demonstrates how soluble braces correspond to soluble solutions of the Yang-Baxter equation.

Contribution

It introduces the notion of solubility for skew left braces and connects it to the solubility of solutions of the Yang-Baxter equation, providing new insights into their structure.

Findings

Soluble skew left braces are characterized by their rich ideal structure.

Soluble non-degenerate set-theoretic solutions correspond to soluble skew left braces.

An example illustrates the relevance of solubility in brace theory.

Abstract

The study of non-degenerate set-theoretic solutions of the Yang-Baxter equation calls for a deep understanding of the algebraic structure of a skew left brace. In this paper, the skew brace theoretical property of solubility is introduced and studied. It leads naturally to the notion of solubility of solutions of the Yang-Baxter equation. It turns out that soluble non-degenerate set-theoretic solutions are characterised by soluble skew left braces. The rich ideal structure of soluble skew left braces is also shown. A worked example showing the relevance of the brace theoretical property of solubility is also presented.