Positive representations & modular functor

Positive representations of quantum groups were defined by I. Frenkel and I. Ip as certain q-deformations of principal series representations of universal enveloping algebras. It was conjectured by the authors that these representations are closed under tensor products. This conjecture happens to be closely related to the modular functor conjecture by V. Fock and A. Goncharov. I will speak about joint works with Gus Schrader (some of which are works in progress) where weprove the above conjectures.