Weak quasi-Hopf algebras and conformal field theory

The aim of my talk is to present results on existence of braided tensor equivalences between braided tensor categories with type A fusion rules from isomprphisms of their Grothendieck rings, and moreover to construct unitary structures on tensor categories arising from affine vertex operator algebras for all Lie types based on Kazhdan-Lusztig theory, Wenzl work and Finkelberg equivalence theorem, assuming that this theorem holds in the setting of Huang-Lepowsky tensor product theory, which is not proven in the literature.