Fusion for the Yang-Baxter equation and the braid group

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.