Positive Representations of Split Real Simply-Laced Quantum Groups

We construct the positive principal series representations for U_q( g_ R) where g is of simply-laced type, parametrized by R≥ 0^r where r is the rank of g . We describe explicitly the actions of the generators in the positive representations as positive essentially self-adjoint operators on a Hilbert space, and prove the transcendental relations between the generators of the modular double. We define the modified quantum group Uq q̃( g_ R) of the modular double and show that the representations of both parts of the modular double commute weakly with each other, there is an embedding into a quantum torus algebra, and the commutant contains its Langlands dual.