The braided group of a square-free solution of the Yang-Baxter equation and its group algebra

Abstract

Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution $(X,r)$ consists of a set $X$ and a bijective map $r:X\times X\to X\times X$ which satisfies the braid relations. In this work we study the braided group $G=G(X,r)$ of an involutive square-free solution $(X,r)$ of finite order $n$ and cyclic index $p=p(X,r)$ and the group algebra $\textbf{k} [G]$ over a field $\textbf{k}$. We show that $G$ contains a $G$-invariant normal subgroup $\mathcal{F}_p$ of finite index $p^n$, $\mathcal{F}_p$ is isomorphic to the free abelian group of rank $n$. We describe explicitly the quotient braided group $\widetilde{G}=G/\mathcal{F}_p$ of order $p^n$ and show that $X$ is embedded in $\widetilde{G}$. We prove that the group algebra $\textbf{k} [G]$ is a free left (resp. right) module of finite rank $p^n$ over its commutative subalgebra $\textbf{k}[\mathcal{F}_p]$ and give an explicit free basis. The center of $\textbf{k} [G]$ contains the subalgebra of symmetric polynomials in $\textbf{k} [x_1^p, \cdots, x_n^p]$. Classical results on group rings imply that $\textbf{k}[G]$ is a left (and right) Noetherian domain of finite global dimension.