Reflections to set-theoretic solutions of the Yang-Baxter equation

TL;DR

This paper investigates reflections in set-theoretic solutions of the Yang-Baxter equation, linking them to derived solutions, and extends existing results to more general cases with computational data.

Contribution

It introduces methods to determine reflections for bijective, non-degenerate solutions and extends results to broader classes of solutions beyond involutive cases.

Findings

Reflections can be studied via derived solutions.

Extended reflection results to non-involutive solutions.

Numerical data on solutions related to small skew braces.

Abstract

The main aim of this paper is to determine reflections to bijective and non-degenerate solutions of the Yang-Baxter equation, by exploring their connections with their derived solutions. This is motivated by a recent description of left non-degenerate solutions in terms of a family of automorphisms of their associated left rack. In some cases, we show that the study of reflections for bijective and non-degenerate solutions can be reduced to those of derived type. Moreover, we extend some results obtained in the literature for reflections of involutive non-degenerate solutions to more arbitrary solutions. Besides, we provide ways for defining reflections for solutions obtained by employing some classical construction techniques of solutions. Finally, we gather some numerical data on reflections for bijective non-degenerate solutions associated with skew braces of small order.