Hecke algebras for set-theoretical solutions to the Yang--Baxter equation

TL;DR

This paper introduces a new Hecke algebra framework for set-theoretical solutions to the Yang--Baxter equation, comparing it to known algebraic structures and exploring specific cases like Torus Knot groups.

Contribution

It defines a novel Hecke algebra for these solutions, analyzes its properties, and relates it to existing algebraic constructions, highlighting key differences from classical Coxeter groups.

Findings

The new Hecke algebra differs from classical Artin--Tits groups.

Identifies distinctions between finite Coxeter groups and Coxeter-like groups.

Connects the algebraic structures to Torus Knot and Complex Reflection groups.

Abstract

We define a concept of Hecke algebra for structure groups of set-theoretical solutions to the Yang--Baxter equation. As a comparison to Artin--Tits groups of spherical type, we study some properties of this construction, while also highlighting some differences that appear, which shows a difference between finite Coxeter groups and the "Coxeter-like" group introduced by Dehornoy. We also relate this definition to known constructions on solutions (retractions). Finally, we study a particular case related to Torus Knot groups and Complex Reflexion groups.