Coprime extensions of indecomposable solutions to the Yang-Baxter equation

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 Cedó and Okniński. 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.