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

TL;DR

This paper characterizes all indecomposable involutive solutions of the Yang-Baxter equation with multipermutation level 2, providing a construction method and classification up to isomorphism, especially for small sizes.

Contribution

It offers a complete classification and construction of indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level 2 with non-abelian permutation groups.

Findings

Constructed a family of solutions of the Yang-Baxter equation.

Proved all such solutions are homomorphic images of the constructed family.

Enumerated solutions of small sizes up to isomorphism.

Abstract

We give a complete characterization of all indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level~2. In the first step we present a construction of some family of such solutions and in the second step we prove that every indecomposable involutive solution of the Yang-Baxter equation with multipermutation level~2 is a homomorphic image of a solution previously constructed. Analyzing this epimorphism, we are able to obtain all such solutions up to isomorphism and enumerate these of small sizes.