A counterpart of the modular double for quantum superalgebra Uq(osp(1\2)) is constructed by means of supersymmetric quantum mechanics. We also construct the R-matrix operator acting in the corresponding representations, which is expressed via quantum dilogarithm.

We study the positive representations Pλ of split real quantum groups Uqq (gℝ) restricted to the Borel subalgebra Uqq (bℝ). We prove that the restriction is independent of the parameter λ. Furthermore, we prove that it can be constructed from the GNS-representation of the multiplier Hopf algebra UqqC ∗ (b ℝ) defined earlier, which allows us to decompose their tensor product using the theory of the “multiplicative unitary”. In particular, the quantum mutation operator can be constructed from the multiplicity module, which will be an essential ingredient in the construction of quantum higher Teichmüller theory from the perspective of representation theory, generalizing earlier work by Frenkel-Kim.

We showed that there is a complete analogue of a representation of the quantum plane ℬq where \q\ = 1, with the classical ax+b group. We showed that the Fourier transform of the representation of ℬ-q$ on ℋ = L2(ℝ) has a limit (in the dual corepresentation) toward the Mellin transform of the unitary representation of the ax+b group, and furthermore the intertwiners of the tensor products representation has a limit toward the intertwiners of the Mellin transform of the classical ax+b representation. We also wrote explicitly the multiplicative unitary defining the quantum ax+b semigroup and showed that it defines the corepresentation that is dual to the representation of ℬq above, and also correspond precisely to the classical family of unitary representation of the ax+b group.

We give complete detail of the description of the GNS representation of the quantum plane A and its dual  as a von Neumann algebra. In particular, we obtain a rather surprising result that the multiplicative unitary W is manageable in this quantum semigroup context. We study the quantum double group construction introduced by Woronowicz, and using Baaj and Vaes' construction of the multiplicative unitary Wm, we give the GNS description of the quantum double D(A) which is equivalent to GLq+(2, ℝ). Furthermore, we study the fundamental corepresentation Tλ, t and its matrix coefficients, and show that it can be expressed by the b-hypergeometric function. We also study the regular corepresentation and representation induced by Wm and prove that the space of L2 functions on the quantum double decomposes into the continuous series representation of Uqq̃(gl(2, ℝ)) with the quantum dilogarithm {pipe}Sb(Q+2iα){pipe}2 as the Plancherel measure. Finally, we describe certain representation theoretic meaning of integral transforms involving the quantum dilogarithm function.

Perspectives in Representation Theory: A Conference in Honor of Igor Frenkel’s 60th Birthday on Perspectives in Representation Theory, Yale University, New Haven, 12-17 May 2012