Fusion for the Yang-Baxter equation and the braid group

TL;DR

This paper explains the fusion procedure for the Yang-Baxter equation, leading to new algebraic structures called fused Hecke algebras, which relate to braid groups and quantum groups.

Contribution

It introduces the fusion method for the Yang-Baxter equation and constructs new fused Hecke algebras, expanding the algebraic framework related to braid groups and quantum groups.

Findings

Introduction of fused Hecke algebras as new solutions to the Yang-Baxter equation

Demonstration of the fusion procedure's role in algebra construction

Connections established between braid groups, quantum groups, and fused algebras

Abstract

These are the extended notes of a mini-course given at the school WinterBraids X. We discuss algebras simultaneously related to: the braid group, the Yang-Baxter equation and the representation theory of quantum groups. The main goal is to explain the idea of the fusion procedure for the Yang-Baxter equation and to show how it leads to new examples of such algebras: the fused Hecke algebras.