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

TL;DR

This paper explores the algebraic structure of groups arising from set-theoretic solutions to the Yang-Baxter equation, demonstrating Garsideness properties and constructing a basis for their algebraic images.

Contribution

It introduces Garsideness properties for structure groups of involutive set-theoretic solutions and constructs a finite basis for their algebraic images.

Findings

Established a homomorphism from the structure group to an algebra.

Constructed a finite basis for the algebraic image using Garsideness.

Extended the analogy with braid groups and Temperley-Lieb algebra.

Abstract

There exists a multiplicative homomorphism from the braid group B to the Temperley-Lieb algebra TL. Moreover, the homomorphic images in TL of the simple elements form a basis for the vector space underlying TL. In analogy with the case of B, there exists a multiplicative homomorphism from the structure group G of a non-degenerate, involutive set-theoretic solution 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 using the Garsideness properties of the solution.