Non-involutive solutions of the Yang-Baxter equation of multipermutation level 2

TL;DR

This paper develops an algorithmic approach to classify and enumerate non-involutive, multipermutation level 2 solutions of the Yang-Baxter equation, expanding understanding of their structure beyond 2-reductive cases.

Contribution

It introduces a method to construct and classify non-2-reductive solutions of the Yang-Baxter equation of level 2, including an enumeration up to size 6.

Findings

Developed an effective construction method for these solutions.

Provided an algorithm to classify solutions up to isomorphism.

Enumerated all solutions of size up to 6.

Abstract

We study non-degenerate set-theoretic solutions of the Yang-Baxter equation of multipermutation level 2 which are not 2-reductive. We describe an effective way of constructing such solutions using square-free 2-reductive solutions and two bijections. We present an algorithm how to obtain all such finite solutions, up to isomorphism. Using this algorithm, we enumerate all solutions of multipermutation level 2 up to size 6.