On various types of nilpotency of the structure monoid and group of a set-theoretic solution of the Yang--Baxter equation

TL;DR

This paper characterizes when the structure monoid of a set-theoretic solution to the Yang--Baxter equation is nilpotent, explores properties of solutions from racks and quandles, and relates multipermutation solutions to group solvability and nilpotency.

Contribution

It provides new characterizations of nilpotency in structure monoids and groups associated with set-theoretic solutions, extending previous results to non-involutive and infinite cases.

Findings

Structure monoid $M(X,r)$ is Malcev nilpotent under certain conditions.

Finite abelian racks and quandles are characterized via these properties.

If $G=G(X,r)$ is nilpotent, its torsion part is finite and related to the commutator subgroup.

Abstract

Given a finite bijective non-degenerate set-theoretic solution $(X,r)$ of the Yang--Baxter equation we characterize when its structure monoid $M(X,r)$ is Malcev nilpotent. Applying this characterization to solutions coming from racks, we rediscover some results obtained recently by Lebed and Mortier, and by Lebed and Vendramin on the description of finite abelian racks and quandles. We also investigate bijective non-degenerate multipermutation (not necessarily finite) solutions $(X,r)$ and show, for example, that this property is equivalent to the solution associated to the structure monoid $M(X,r)$ (respectively structure group $G(X,r)$) being a multipermuation solution and that $G=G(X,r)$ is solvable of derived length not exceeding the multipermutation level of $(X,r)$ enlarged by one, generalizing results of Gateva-Ivanova and Cameron obtained in the involutive case. Moreover, we also prove that if $X$ is finite and $G=G(X,r)$ is nilpotent, then the torsion part of the group $G$ is finite, it coincides with the commutator subgroup $[G,G]_+$ of the additive structure of the skew left brace $G$ and $G/[G,G]_+$ is a trivial left brace.