On some Garsideness properties of structure groups of set-theoretic solutions of the Yang-Baxter equation

Abstract

There exists a multiplicative homomorphism from the braid group on k + 1 strands to the Temperley–Lieb algebra TLk . Moreover, the homomorphic images in TLk of the simple elements form a basis for the vector space underlying TLk . In analogy with the case of Bk , there exists a multiplicative homomorphism from the structure group G(X, r) of a non-degenerate, involutive set-theoretic solution (X, r), with , to an algebra, which extends to a homomorphism of algebras. We construct a finite basis of the underlying vector space of the image of G(X, r) using the Garsideness properties of G(X, r).