On derived-indecomposable solutions of the Yang--Baxter equation

TL;DR

This paper investigates the structure of solutions to the Yang--Baxter equation using skew braces, revealing their torsion, radical, nilpotency, and finite generation properties, especially when solutions are indecomposable.

Contribution

It introduces a brace-theoretic analogue of $FC$-groups and explores their fundamental properties and connections to YBE solutions, especially in the indecomposable case.

Findings

Skew braces associated with solutions are $FC$-groups with finitely many conjugates.

The multiplicative group of these braces is virtually abelian.

The paper establishes torsion, radical, nilpotency, and finite generation properties for these braces.

Abstract

If $(X,r)$ is a finite non-degenerate set-theoretic solution of the Yang--Baxter equation, the additive group of the structure skew brace $G(X,r)$ is an $FC$-group, i.e. a group whose elements have finitely many conjugates. Moreover, its multiplicative group is virtually abelian, so it is also close to an $FC$-group itself. If one additionally assumes that the derived solution of $(X,r)$ is indecomposable, then for every element $b$ of $G(X,r)$ there are finitely many elements of the form $b*c$ and $c*b$, with $c\in G(X,r)$. This naturally leads to the study of a brace-theoretic analogue of the class of $FC$-groups. For this class of skew braces, the fundamental results and their connections with the solutions of the YBE are described: we prove that they have good torsion and radical theories and they behave well with respect to certain nilpotency concepts and finite generation.