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

TL;DR

This paper explores the deep connections between braces, symmetric groups, and set-theoretic solutions of the Yang-Baxter equation, introducing new invariants and criteria for classifying solutions and their algebraic structures.

Contribution

It establishes a novel relationship between symmetric groups and braces, introduces the derived chain of ideals as an invariant, and provides new criteria for identifying multipermutation solutions.

Findings

Symmetric groups of finite multipermutation level are solvable with bounded length.

Introduces the derived chain of ideals as a new invariant for symmetric groups.

Provides explicit criteria for recognizing multipermutation solutions.

Abstract

We involve simultaneously the theory of matched pairs of groups and the theory of braces to study set-theoretic solutions of the Yang-Baxter equation (YBE). We show the intimate relation between the notions of a symmetric group (a braided involutive group) and a left brace, and find new results on symmetric groups of finite multipermutation level and the corresponding braces. We introduce a new invariant of a symmetric group $(G,r)$, \emph{the derived chain of ideals of} $G$, which gives a precise information about the recursive process of retraction of $G$. We prove that every symmetric group $(G,r)$ of finite multipermutation level $m$ is a solvable group of solvable length at most $m$. To each set-theoretic solution $(X,r)$ of YBE we associate two invariant sequences of symmetric groups: (i) the sequence of its derived symmetric groups; (ii) the sequence of its derived permutation groups and explore these for explicit descriptions of the recursive process of retraction. We find new criteria necessary and sufficient to claim that $(X, r)$ is a multipermutation solution.