Indecomposable set-theoretical solutions to the Quantum Yang-Baxter Equation on a set with prime number of elements

TL;DR

This paper classifies all indecomposable nondegenerate set-theoretical solutions to the Quantum Yang-Baxter equation on prime-sized sets, showing they are affine and providing a complete classification.

Contribution

It proves that such solutions are affine and offers a simple classification, extending previous involutive case results and utilizing advanced group-theoretical methods.

Findings

All indecomposable solutions are affine.

Complete classification of solutions on prime sets.

Reduction to a group-theoretical problem.

Abstract

In this paper we show that all indecomposable nondegenerate set-theoretical solutions to the Quantum Yang-Baxter equation on a set of prime order are affine, which allows us to give a complete and very simple classification of such solutions. This result is a natural application of the general theory of set-theoretical solutions to the quantum Yang-Baxter equation. It is also a generalization of the corresponding statement for involutive set-theoretical solutions proved in an earlier paper of P.E. and A.S. with T.Schedler. In order to prove our main result, we use the classification theory developed by the third author (based on the ideas of Lu, Yan, and Zhu) to reduce the problem to a group-theoretical statement: a finite group with trivial center generated by a conjugacy class of prime order is a subgroup of the affine group. Unfortunalely, we did not find an elementary proof of this statement, and our proof relies on the classification of outer automorphisms of finite simple groups.