Garside groups and Yang-Baxter equation

TL;DR

This paper explores the relationship between Garside groups and solutions to the quantum Yang-Baxter equation, providing a classification framework and characterizations for indecomposable solutions.

Contribution

It establishes a one-to-one correspondence between certain Garside groups and set-theoretical solutions of the Yang-Baxter equation, and characterizes indecomposable solutions via $ riangle$-pure Garside groups.

Findings

Correspondence between Garside groups and Yang-Baxter solutions

Characterization of indecomposable solutions

Structural insights into Garside groups and solutions

Abstract

We establish a one-to-one correspondence between a class of Garside groups admitting a certain presentation and the structure groups of non-degenerate, involutive and braided set-theoretical solutions of the quantum Yang-Baxter equation. We also characterize indecomposable solutions in terms of $\Delta$-pure Garside groups.