Studying solutions of the Yang-Baxter equation through skew braces, with an application to indecomposable involutive solutions with abelian permutation group

TL;DR

This paper explores the relationship between set-theoretic solutions to the Yang-Baxter equation and permutation skew braces, introducing new invariants, analyzing automorphisms, and classifying solutions with abelian permutation groups.

Contribution

It introduces a variation of the multipermutation level for solutions, shows its equivalence to the skew brace level, and classifies finite indecomposable involutive solutions with abelian permutation groups.

Findings

The new multipermutation level coincides with that of the permutation skew brace.

Automorphism groups of solutions are characterized via their permutation skew braces.

Complete classification of finite indecomposable involutive solutions with abelian permutation groups.

Abstract

We connect properties of set-theoretic solutions to the Yang--Baxter equation to properties of their permutation skew brace. In particular, a variation of the multipermutation level of a solution is presented and we show that it coincides with the multipermutation level of the permutation skew brace, contrary to the inequality that one has for the usual multipermutation level of solutions. We relate the number of orbits of a solution to generators of its permutation skew brace and relate different kinds of notions of generating sets of a skew brace. Also, the automorphism groups of solutions are studied through their permutation skew brace. As an application, we obtain a surprising result on subsolutions of multipermutation solutions and we give a description of all finite indecomposable involutive solutions to the Yang--Baxter equation with abelian permutation group. For multipermutation level 3, we obtain the precise number of isomorphism classes of such solutions of a given size.