The braided Doplicher-Roberts program and the Finkelberg-Kazhdan-Lusztig equivalence: A historical perspective, recent progress, and future directions

Our recent approach to the Finkelberg-Kazhdan-Lusztig equivalence theorem centers on the construction of a fiber functor associated with the categories in the equivalence theorem, which in turn explains the underlying algebraic and analytic structure of the corresponding weak Hopf algebra in a new sense. We provide a non-technical and historical overview of the core arguments behind our proof, discuss these structural properties, and its applications to rigidity and unitarizability of braided fusion categories arising from conformal field theory. We conclude proposing some natural directions for future research.