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

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.

15 pages