Coprime extensions of indecomposable solutionsto the Yang–Baxter equation

Abstract

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 and 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.