Simplicity and finite primitive level of indecomposable set-theoretic solutions of the Yang-Baxter equation

TL;DR

This paper explores finite indecomposable set-theoretic solutions to the Yang-Baxter equation, focusing on simplicity and primitive level, providing group-theoretic characterizations and addressing open questions.

Contribution

It introduces new group-theoretic characterizations of simple and primitive level solutions, advancing the understanding of their structure within the Yang-Baxter framework.

Findings

Characterization of simple solutions via permutation groups

Description of solutions with finite primitive level

Addressing open questions in the theory

Abstract

This paper aims to deepen the theory of bijective non-degenerate set-theoretic solutions of the Yang-Baxter equation, not necessarily involutive, by means of q-cycle sets. We entirely focus on the finite indecomposable ones among which we especially study two classes of current interest: the simple solutions and those having finite primitive level. In particular, we provide two group-theoretic characterizations of these solutions, involving their permutation groups. Finally, we deal with some open questions.