  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.
