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

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.