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 \(r:X\times X\rightarrow X\times X\) 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 \(\mathbf{k} [G]\) over a field \(\mathbf{k} \). We show that G contains an invariant subgroup \({\mathcal {F}}_p\) of finite index \(p^n\) which 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 \(\mathbf{k} [G]\) is a free left (resp. right) module of finite rank \(p^n\) over its commutative subalgebra \(\mathbf{k} [{\mathcal {F}}_p]\) and give an explicit free basis. The center of \(\mathbf{k} [G]\) contains the subalgebra of symmetric polynomials in \(\mathbf{k} [x_1^p, \cdots , x_n^p]\). Classical results on group rings imply that \(\mathbf{k} [G]\) is a left (and right) Noetherian domain of finite global dimension.