Positive Representations of Split Real Quantum Group

In this talk, I will present a construction of a family of positive principal series representations for the modular double of Uq(sl(n,R)) by positive self adjoint operators, using a parameterization by cluster variables. This is a generalization of the special class of representations considered by J. Teschner in n=2, where it exhibits a strong parallel to the finite-dimensional representation theory of quantum groups. I will discuss some of these properties, and conjecture their generalizations to the family of positive representations in the higher rank.