We found an explicit construction of a representation of the positive quantum group and its modular double by positive essentially self-adjoint operators. Generalizing Lusztig's parametrization, we found a Gauss type decomposition for the totally positive quantum group parametrized by the standard decomposition of the longest element . Under this parametrization, we found explicitly the relations between the standard quantum variables, the relations between the quantum cluster variables, and realizing them using non-compact generators of the q-tori by positive essentially self-adjoint operators. The modular double arises naturally from the transcendental relations, and an space in the von Neumann setting can also be defined.

We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for C a braided tensor category, we give a detailed construction of the canonical algebra in C⊠C^rev: if C is semisimple but not necessarily finite or rigid, then ⨁X∈Irr(C) X′⊠X is a commutative algebra, with X′ a representing object for Hom_C(∙⊗_C X,1_C). Conversely, let A=⨁i∈I U_i ⊠ V_i be a simple commutative algebra in U⊠V with U semisimple and rigid but not necessarily finite, and V rigid but not necessarily semisimple. If the unit objects of U and V form a commuting pair in A, we show there is a braid-reversed equivalence between subcategories of U and V sending U_i to (V_i)*. When U and V are module categories for simple vertex operator algebras U and V, we glue U and V along U⊠V via a map τ: Irr(U)→Obj(V) such that τ(U)=V to create A=⨁X∈Irr(U) X′⊗τ(X). Thus under certain conditions, τ extends to a braid-reversed equivalence between U and V if and only if A is a simple conformal vertex algebra extending U⊗V. As examples, we glue Kazhdan-Lusztig categories at generic levels to obtain new vertex algebras extending the tensor product of two affine vertex algebras, and we prove braid-reversed equivalences between certain module categories for affine vertex algebras and W-algebras at admissible levels.

We investigate compact quantum group actions on unital $C^*$-algebras by analyzing invariant subsets and invariant states. In particular, we come up with the concept of compact quantum group orbits and use it to show that countable compact metrizable spaces with infinitely many points are not quantum homogeneous spaces.

We find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra g^, from g-module homomorphisms. When g=sl_2, these results extend previous joint work with J. Yang, but the method used here is different. Here, we construct intertwining operators by solving Knizhnik-Zamolodchikov equations for three-point correlation functions associated to g^, and we identify obstructions to the construction arising from the possible non-existence of series solutions having a prescribed form.

We describe the generalized Casimir operators and their actions on the positive representations Pλ of the modular double of split real quantum groups Uqq (gR). We introduce the notion of virtual highest and lowest weights, and show that the central characters admit positive values for all parameters λ. We show that their image defines a semi-algebraic region bounded by real points of the discriminant variety independent of q, and we discuss explicit examples in the lower rank cases.