"Set-theoretical" solutions of the quantum Yang-Baxter equation and a class of Garside groups

TL;DR

This paper explores the relationship between set-theoretical solutions to the quantum Yang-Baxter equation and Garside groups, revealing a correspondence that characterizes indecomposable solutions via Garside group properties.

Contribution

It establishes a one-to-one correspondence between certain set-theoretical solutions of the quantum Yang-Baxter equation and Garside groups with specific presentations, and characterizes indecomposability.

Findings

Correspondence between solutions and Garside groups

Indecomposability characterized by Δ-pure Garside groups

Provides a new framework for understanding quantum Yang-Baxter solutions

Abstract

We establish a one-to-one correspondence between structure groups of non-degenerate, involutive and braided "set-theoretical" solutions of the quantum Yang-Baxter equation and Garside groups with a certain presentation. Moreover, we show that the solution is indecomposable if and only if its structure group is a $\Delta-$pure Garside group.