Indecomposable involutive solutions of the Yang-Baxter equation of multipermutational level 2 with abelian permutation group

TL;DR

This paper classifies all finite indecomposable involutive solutions of the Yang-Baxter equation with multipermutational level at most 2 and abelian permutation groups, providing formulas and properties of their automorphism groups.

Contribution

It offers a complete construction and enumeration of such solutions, and characterizes their automorphism groups as regular abelian groups.

Findings

All such solutions are constructed explicitly.

A formula for counting solutions with a fixed number of elements is derived.

Automorphism groups are shown to be regular abelian groups.

Abstract

We present a construction of all finite indecomposable involutive solutions of the Yang-Baxter equation of multipermutational level at most 2 with abelian permutation group. As a consequence, we obtain a formula for the number of such solutions with a fixed number of elements. We also describe some properties of the automorphism groups in this case - in particular, we show they are regular abelian groups.