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

Edouard Feingesicht

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.

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

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.

Paper Structure

This paper contains 8 sections, 35 theorems, 39 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Cycle sets
  3. Braces
  4. Defining the Hecke algebra
  5. Anti-involution on the Hecke algebra
  6. Semi-simplicity
  7. Two-generated Cyclic group
  8. Finding the correct definition via a diagrammtic approach

Key Result

Theorem 1

Let $(X,r)$ be an involutive non-degenerate set-theoretical solution to the Yang--Baxter equation of size $n$ and Dehornoy's class $d$. Denote $G$ the structure group of $(X,r)$ and $\overline G_2=G/\langle x^{[2d]}\rangle_{x\in X}$ its germ associated to $2d$ (two times Dehornoy's class). For any i Then the followings hold:

Figures (1)

  • Figure :

Theorems & Definitions (88)

  • Theorem
  • Definition 1.1: rump
  • Definition 1.2: rump
  • Example 1.3
  • Remark 1.4
  • Definition 1.5: rump07brace
  • Remark 1.6
  • Example 1.7
  • Example 1.9
  • Lemma 1.10: brace
  • ...and 78 more