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

Fusion for the Yang-Baxter equation and the braid group