Unitary modular fusion categories arise in various frameworks. After a general overview on unitarity, we discuss those with Verlinde fusion rules arising from quantum groups at roots of unity. The notion of weak quasi-Hopf algebra was introduced by Mack and Schomerus as an extension of the notion of quasi-Hopf algebra of Drinfeld. We shall associate semisimple weak quasi-Hopf algebras to these above fusion categories following ideas of Wenzl for all Lie types. We discuss a conjectural application to a direct proof of Kazhdan-Lusztig-Finkelberg theorem.
My talk is based on a joint work with S. Carpi, S. Ciamprone, M.V. Giannone