Solutions of the Yang-Baxter equation associated to skew left braces, with applications to racks

TL;DR

This paper introduces a method to construct all non-degenerate set-theoretic solutions to the Yang-Baxter equation associated with a given skew left brace, enabling the recovery of racks from permutation groups.

Contribution

It provides a comprehensive construction method linking skew left braces to solutions of the Yang-Baxter equation, including special properties and rack recovery.

Findings

Constructed all solutions with a given skew left brace

Extended method to solutions with properties like involutive or square-free

Demonstrated how to recover racks from permutation groups

Abstract

Given a skew left brace $B$, a method is given to construct all the non-degenerate set-theoretic solutions $(X,r)$ of the Yang Baxter equation such that the associated permutation group $\mathcal{G}(X,r)$ is isomorphic, as a skew left brace, to $B$. This method depends entirely on the brace structure of $B$. We then adapt this method to show how to construct solutions with additional properties, like square-free, involutive or irretractable solutions. Using this result, it is even possible to recover racks from their permutation group.