We study irreducible spherical unitary representations of the Drinfeld double of the q-deformation of a connected simply connected compact Lie group, which can be considered as a quantum analogue of the complexification of the Lie group. In the case of SUq (3), we give a complete classification of such representations. As an application, we show the Drinfeld double of the quantum group SUq(2n + 1) has property (T), which also implies central property (T) of the dual of SUq(2n + 1).Definition 2.1. The quantized enveloping algebra U q (g) is the algebra defined by generators {K λ , E α , F α | λ ∈ P, α ∈ Π} and relations
The classical Gaussian functor associates to every orthogonal representation of a locally compact group G a probability measure preserving action of G called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of G on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine isometric actions. We use the Patterson-Sullivan theory as well as Lyons-Pemantle work on tree-indexed random walks in order to give a precise description of this phase transition for affine isometric actions of groups acting on trees. Finally, we use Gaussian actions to show that every nonamenable locally compact group without property (T) admits a free nonamenable weakly mixing action of stable type III1.
The classical Gaussian functor associates to every orthogonal representation of a locally compact group G a probability measure preserving action of G called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of G on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine isometric actions. We use the Patterson-Sullivan theory as well as Lyons-Pemantle work on tree-indexed random walks in order to give a precise description of this phase transition for affine isometric actions of groups acting on trees. We also show that every locally compact group without property (T) admits a nonsingular Gaussian that is free, weakly mixing and of stable type III 1 .
Torsion-freeness for discrete quantum groups was introduced by R. Meyer in order to formulate a version of the Baum-Connes conjecture for discrete quantum groups. In this note, we introduce torsionfreeness for abstract fusion rings. We show that a discrete quantum group is torsion-free if its associated fusion ring is torsion-free. In the latter case, we say that the discrete quantum group is strongly torsionfree. As applications, we show that the discrete quantum group duals of the free unitary quantum groups are strongly torsion-free, and that torsion-freeness of discrete quantum groups is preserved under Cartesian and free products. We also discuss torsion-freeness in the more general setting of abstract rigid tensor C * -categories.
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