Left semi-braces and solutions to the Yang-Baxter equation

TL;DR

This paper extends algebraic and structural results related to solutions of the Yang-Baxter equation from non-degenerate cases to more general solutions associated with left semi-braces, including degenerate and idempotent solutions.

Contribution

It introduces a new framework for analyzing solutions to the Yang-Baxter equation via left semi-braces, broadening the scope beyond non-degenerate solutions.

Findings

Describes semi-braces and their properties

Proves decomposition results for semi-braces

Extends algebraic properties to broader classes of solutions

Abstract

Let $r:X^{2}\rightarrow X^{2}$ be a set-theoretic solution of the Yang-Baxter equation on a finite set $X$. It was proven by Gateva-Ivanova and Van den Bergh that if $r$ is non-degenerate and involutive then the algebra $K\langle x \in X \mid xy =uv \mbox{ if } r(x,y)=(u,v)\rangle$ shares many properties with commutative polynomial algebras in finitely many variables; in particular this algebra is Noetherian, satisfies a polynomial identity and has Gelfand-Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non-degenerate bijective solutions. Such solutions are naturally associated to finite skew left braces. In this paper we will prove an analogue result for arbitrary solutions $r_B$ that are associated to a left semi-brace $B$; such solutions can be degenerate or can even be idempotent. In order to do so we first describe such semi-braces and we prove some decompositions results extending results of Catino, Colazzo, and Stefanelli.