Solutions of the Yang-Baxter equation associated with a left brace

TL;DR

This paper presents a method to construct all involutive, non-degenerate set-theoretic solutions to the Yang-Baxter equation associated with a given left brace, leveraging the brace's algebraic structure.

Contribution

It introduces a comprehensive construction method for solutions of the YBE based on the structure of left braces, connecting algebraic properties to solutions.

Findings

Provides a complete construction method for YBE solutions from left braces

Establishes a correspondence between brace structure and YBE solutions

Enables classification of solutions based on brace properties

Abstract

Given a left brace $G$, a method to construct all the involutive, non-degenerate set-theoretic solutions $(Y,s)$ of the YBE, such that $\mathcal{G}(Y,s)\cong G$ is given. This method depends entirely on the brace structure of $G$.