Abelian maps, brace blocks, and solutions to the {Y}ang-{B}axter equation

TL;DR

This paper explores how endomorphisms with abelian images in nonabelian groups generate structures called brace blocks, leading to numerous solutions to the Yang-Baxter equation, expanding the understanding of algebraic solutions in this context.

Contribution

It introduces a novel construction linking abelian-image endomorphisms to brace blocks and solutions to the Yang-Baxter equation, providing new methods to generate solutions.

Findings

Endomorphisms with abelian images induce families of binary operations.

Brace blocks produce many non-degenerate solutions to the Yang-Baxter equation.

Number of solutions can be arbitrarily large.

Abstract

Let $G$ be a finite nonabelian group. We show how an endomorphism of $G$ with abelian image gives rise to a family of binary operations $\{\circ_n: n\in \mathbb Z^{\ge 0}\}$ on $G$ such that $(G,\circ_m,\circ_n)$ is a skew left brace for all $m,n\ge 0$. A brace block gives rise to a number of non-degenerate set-theoretic solutions to the Yang-Baxter equation. We give examples showing that the number of solutions obtained can be arbitrarily large.