Abelian fixed point free endomorphisms and the Yang-Baxter equation

TL;DR

This paper introduces a simple family of solutions to the set-theoretic Yang-Baxter equation derived from special endomorphisms of finite groups, with explicit examples from various group classes.

Contribution

It presents a novel method to generate solutions to the Yang-Baxter equation using abelian fixed point free endomorphisms of finite groups.

Findings

Constructed solutions are inverse to each other.

Explicit examples with dihedral, alternating, symmetric, and metacyclic groups.

Provided a new link between group endomorphisms and Yang-Baxter solutions.

Abstract

We obtain a simple family of solutions to the set-theoretic Yang-Baxter equation, one which depends only on considering special endomorphisms of a finite group. We show how such an endomorphism gives rise to two non-degenerate solutions to the Yang-Baxter equation, solutions which are inverse to each other. We give concrete examples using dihedral, alternating, symmetric, and metacyclic groups.