Positive representations of quantum groups and higher Teichmuller theory

Abstract

Positive representations are infinite-dimensional bimodules for the quantum group and its modular dual where both act by positive essentially self-adjoint operators. Fifteen years ago Ponsot and Teschner showed that positive representations are closed under taking tensor products in the case g = sl(2), however similar conjecture remains open for all other types. I will outline its proof for g = sl(n) based on a joint work in progress with Gus Schrader. I will also argue that this conjecture is the key step towards the proof of the modular functor conjecture for quantized higher Teichmuller theories.