Finite idempotent set-theoretic solutions of the Yang--Baxter equation

TL;DR

This paper characterizes finite idempotent set-theoretic solutions to the Yang-Baxter equation using semigroup structures and explores the algebraic properties of their associated structure algebras.

Contribution

It provides a complete description of finite idempotent solutions via semigroup structures and characterizes when their structure monoids form groups, along with algebraic properties.

Findings

Finite solutions are determined by a left simple semigroup structure.

The structure algebra is right Noetherian and semiprime in characteristic zero.

The structure semigroup decomposes into cancellative semigroups with specific group quotients.

Abstract

It is proven that finite idempotent left non-degenerate set-theoretic solutions $(X,r)$ of the Yang-Baxter equation on a set $X$ are determined by a left simple semigroup structure on $X$ (in particular, a finite union of isomorphic copies of a group) and some maps $q$ and $\varphi_x$ on $X$, for $x\in X$. This structure turns out to be a group precisely when the associated structure monoid is cancellative and all the maps $\varphi_x$ are equal to an automorphism of this group. Equivalently, the structure algebra $K[M(X,r)]$ is right Noetherian, or in characteristic zero it has to be semiprime. The structure algebra always is a left Noetherian representable algebra of Gelfand--Kirillov dimension one. To prove these results it is shown that the structure semigroup $S(X,r)$ has a decomposition in finitely many cancellative semigroups $S_u$ indexed by the diagonal, each $S_u$ has a group of quotients $G_u$ that is finite-by-(infinite cyclic) and the union of these groups carries the structure of a left simple semigroup. The case that $X$ equals the diagonal is fully described by a single permutation on $X$.