On the indecomposable involutive set-theoretic solutions of the Yang-Baxter equation of prime-power size

TL;DR

This paper presents a comprehensive method for constructing and classifying indecomposable involutive set-theoretic solutions to the Yang-Baxter equation, focusing on prime-power sizes and specific permutation group structures.

Contribution

It introduces a new construction method for all such solutions with prime-power size and cyclic permutation groups, and classifies solutions with abelian permutation groups of size pq.

Findings

Constructed all indecomposable solutions with prime-power size and cyclic permutation groups.

Classified indecomposable solutions with abelian permutation groups of size pq.

Provided a complete framework for understanding these solutions in the specified cases.

Abstract

We develop a method to construct all the indecomposable involutive set-theoretic solutions of the Yang-Baxter equation with a prime-power number of elements and cyclic permutation group. Moreover, we give a complete classification of the indecomposable ones having abelian permutation group and cardinality $pq$ (where $p$ and $q$ are prime numbers not necessarily distinct).