Discrete Quantum Groups Research Articles

K-amenability for amalgamated free products of amenable discrete quantum groups

The basic notions and results of equivariant KK-theory concerning crossed products can be extended to the case of locally compact quantum groups. We recall these constructions and prove some useful properties of subgroups and amalgamated free products of discrete quantum groups. Using these properties and a quantum analogue of the Bass-Serre tree, we establish the K-amenability of amalgamated free products of amenable discrete quantum groups.

The Martin boundary of a discrete quantum group

We consider the Markov operator Pφ on a discrete quantum group given by convolution with a q-tracial state φ. In the study of harmonic elements x, Pφ(x) = x, we define the Martin boundary Aφ. It is a separable C ∗-algebra carrying canonical actions of the quantum group and its dual. We establish a representation theorem to the effect that positive harmonic elements correspond to positive linear functionals on Aφ. The C ∗-algebraAφ has a natural time evolution, and the unit can always be represented by a KMS state. Any such state gives rise to a u.c.p. map from the von Neumann closure of Aφ in its GNS representation to the von Neumann algebra of bounded harmonic elements, which is an analogue of the Poisson integral. Under additional assumptions this map is an isomorphism which respects the actions of the quantum group and its dual. Next we apply these results to identify the Martin boundary of the dual of SUq(2) with the quantum homogeneous sphere of Podleś. This result extends and unifies previous results by Ph. Biane and M. Izumi.

A Complete Formulation of the Baum–Connes Conjecture for the Action of Discrete Quantum Groups

A Complete Formulation of the Baum–Connes Conjecture for the Action of Discrete Quantum Groups

An Algebraic Framework for Group Duality

A Hopf algebra is a pair (A,Δ) whereAis an associative algebra with identity andΔa homomorphism formAtoA⊗Asatisfying certain conditions. If we drop the assumption thatAhas an identity and if we allowΔto have values in the so-called multiplier algebraM(A⊗A), we get a natural extension of the notion of a Hopf algebra. We call this a multiplier Hopf algebra. The motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group. Also for these multiplier Hopf algebras, there is a natural notion of left and right invariance for linear functionals (called integrals in Hopf algebra theory). We show that, if such invariant functionals exist, they are unique (up to a scalar) and faithful. For a regular multiplier Hopf algebra (A,Δ) (i.e., with invertible antipode) with invariant functionals, we construct, in a canonical way, the dual (Â,Δ). It is again a regular multiplier Hopf algebra with invariant functionals. It is also shown that the dual of (Â,Δ) is canonically isomorphic with the original multiplier Hopf algebra (A,Δ). It is possible to generalize many aspects of abstract harmonic analysis here. One can define the Fourier transform; one can prove Plancherel's formula. Because any finite-dimensional Hopf algebra is a regular multiplier Hopf algebra and has invariant functionals, our duality theorem applies to all finite-dimensional Hopf algebras. Then it coincides with the usual duality for such Hopf algebras. But our category of multiplier Hopf algebras also includes, in a certain way, the discrete (quantum) groups and the compact (quantum) groups. Our duality includes the duality between discrete quantum groups and compact quantum groups. In particular, it includes the duality between compact abelian groups and discrete abelian groups. One of the nice features of our theory is that we have an extension of this duality to the non-abelian case, but within one category. This is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type. We prove that also these are dual to each other. We treat an example that is sufficiently general to illustrate most of the different features of our theory. It is also possible to construct the quantum double of Drinfel'd within this category. This provides a still wider class of examples. So, we obtain many more than just the compact and discrete quantum within this setting.

Cosemisimple Bialgebras and Discrete Quantum Semigroups

We show for a cosemisimple bialgebra that a standard *-operation making it into a discrete quantum semigroup must be unique. It may not exist: we prove such an operation on a cosemisimple Oq(SL2(C)) exists if and only if the parameterqis real. We also conclude that discrete quantum groups form a more restrictive class than cosemisimple *-Hopf algebras.

Multiplier Hopf algebras and duality

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

Discrete Quantum Groups

LetGbe any discrete group. Consider the algebraAof all complex functions with finite support onGwith pointwise operations. The multiplication onGinduces a comultiplication Δ onAby (Δf)(p,q)=f(pq) wheneverf∈Aandp,q∈G. IfGis finite, one can identify the algebra of complex functions onG×GwithA⊗Aso that Δ:A→A⊗A. Then (A,Δ) is a Hopf algebra. IfGis infinite, we still have Δ(f)(g⊗1) and Δ(f)(1⊗g) inA⊗Afor allfandg. In this case (A,Δ) is a multiplier Hopf algebra. In fact, it is a multiplier Hopf *-algebra whenAis given the natural involution defined by[formula]for allf∈Aandp∈G. In this paper we call a multiplier Hopf *-algebra (A,Δ)adiscrete quantum group if the underlying *-algebraAis a direct sum of full matrix algebras. We study these discrete quantum groups and we give a simple proof of the existence and uniqueness of a left and a right invariant Haar measure.