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 $B_{k+1}$ on k + 1 strands to the Temperley–Lieb algebra TL_k. Moreover, the homomorphic images in TL_k of the simple elements form a basis for the vector space underlying TL_k. In analogy with the case of B_k, there exists a multiplicative homomorphism from the structure group G(X, r) of a non-degenerate, involutive set-theoretic solution (X, r), with $\left|X\right|=n$, 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).

Keywords:

• Coxeter-like quotient groups
• Garside groups and monoids
• set-theoretic solutions of the quantum Yang-Baxter equation

2020 Mathematics Subject Classification:

• 20F36
• 20F65

Disclosure statement

The author reports there are no competing interests to declare.