Coprime extensions of indecomposable solutions to the Yang-Baxter equation

TL;DR

This paper introduces a new method for extending solutions to the Yang-Baxter equation using equivariant mappings, focusing on coprime extensions and their structure, especially for solutions of size pqr where p, q, r are distinct primes.

Contribution

It develops a framework for coprime extensions of indecomposable solutions to the Yang-Baxter equation using twisted extensions and applies it to classify solutions of size pqr.

Findings

Every coprime extension of indecomposable solutions can be realized as a twisted extension.

Complete description of indecomposable solutions of size pqr for distinct primes.

Connections made to cohomology theory for solutions and cycle set language.

Abstract

In this article, we introduce a method to extend involutive nondegenerate set-theoretic solutions to the Yang--Baxter equation by means of equivariant mappings to graded modules, thus leading to the notion of a twisted extension. Furthermore, we define coprime extensions of solutions and prove that each coprime extension of indecomposable solutions can be obtained as a suitable twisted extension. We then apply our results to obtain a full description of indecomposable solutions of size $pqr$, where $p,q,r$ are different primes, from a structure theorem of Ced\'o and Okni\'nski. We close with some remarks on a cohomology theory for solutions developed by Lebed and Vendramin. We express our results in the language of cycle sets.