Coprime extensions of indecomposable solutions to the Yang-Baxter equation
Carsten Dietzel
TL;DR
The paper develops a coprime-extension framework for indecomposable cycle sets by proving that every coprime extension is equivalent to a twisted extension $X \otimes_{\Phi} A$ with an $X$-graded $\mathcal{G}(X)$-module $A$ and an equivariant map $\Phi$; this yields a concrete description of the permutation group as a semidirect product and clarifies when the extension remains indecomposable. The authors apply the framework to obtain a full description of indecomposable cycle sets of size $pqr$ (distinct primes) by leveraging the Cedó–Okniński structure theorem, giving explicit data-driven constructions (including parallel and non-parallel twists) and noting the auxiliary divisibility condition $q \mid r-1$ in one case. They also connect the approach to Lebed–Vendramin cohomology, showing that under the coprimality assumption twisted $2$-cocycles reduce to equivariant ones, and propose a broader twisted cohomology viewpoint. Overall, the work provides a systematic, module-theoretic toolkit for constructing and classifying coprime extensions of indecomposable cycle sets and demonstrates its effectiveness in the $|X|=pqr$ regime.
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.