Weak quasi Hopf algebras, tensor C∗-categories and conformal field theory

Hopf algebras and their generalisations are often understood as useful ‘linearizations’ of semisimple tensor categories. Two most studied classes are: quasi-Hopf algebras (Drinfeld, 90s) weak Hopf algebras (Bohm, Nill, Szlachanyi, 00s). The Aim of my talk is to: Review of general theory of weak quasi Hopf algebras extending QHA (Drinfeld, Majid, Mack-Schomerus, Haring-Oldenburg). Introduce a special subclass (w-Hopf algebras) with ‘trivial’ associator. Examples include A(slN,l) constructed in 2015 ‘orthogonal’ to SUq(N) for q > 0. Give New examples. Construction of semisimple f.d. weak quasi Hopf C∗-algebras A(g, l), etc, associated to quantum group/ VOA/conformal net fusion categories under certain conditions. Application to construction of tensor C∗-categories associated to the level k affine vertex operator algebras Vgk