Set-theoretic solutions of the Yang-Baxter equation, Braces, and Symmetric groups

Abstract

Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution $(X,r)$ consists of a set $X$ and a bijective map $r:X\times X\to X\times X$ which satisfies the braid relations. In this work we suggest to involve simultaneously the matched pairs of groups theory and the theory of braces to study set-theoretic solutions of YBE. We find new results on symmetric groups of finite multipermutation level and the corresponding braces. We show that a square-free solution $(X,r)$ has finite multipermutation level $m$ \emph{iff} the associated symmetric group $G(X,r)$ does so. We prove that each solutions $(X,r)$ whose associated brace $(G, +,.)$ is a two-sided brace must be a trivial solution. We find new criteria sufficient to claim "$\mpl X < \infty$". We prove that each finite square-free symmetric set $(X,r)$ whose permutation group $\mathcal{G}(X,r)$ has mutually inverse left and right actions (condition \textbf{lri}) is a multipermutation solution.