Involutive (simple) latin solutions of the Yang-Baxter equation and related (left) quasigroups

TL;DR

This paper classifies involutive set-theoretic solutions to the Yang-Baxter equation, focusing on their structure, simplicity, and permutation groups, with detailed descriptions when the displacement group is abelian.

Contribution

It provides a complete classification of certain involutive solutions, characterizes simple solutions with nilpotent groups, and connects these solutions to algebraic structures like the Weyl algebra.

Findings

Classified blocks of imprimitivity and congruences of irretractable solutions.

Characterized simple solutions with nilpotent permutation groups.

Enumerated simple solutions of minimal size p^p for prime p.

Abstract

In this paper, we study involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation with regular displacement group. In particular, we completely describe the blocks of imprimitivity and the congruences of the irretractable ones, that we show belonging to the class of the latin solutions. Among these solutions, we characterise the simple ones having nilpotent permutation group. A more precise description involving the First Weyl Algebra will be provided when the displacement group is abelian and normal in the total permutation group, and we enumerate and classify the simple ones having minimal size $p^p$, for an arbitrary prime number $p$. Finally, we illustrate our results by some examples.