On projective representations for compact quantum groups

Abstract

We study actions of compact quantum groups on type I -factors, which may be interpreted as projective representations of compact quantum groups. We generalize to this setting some of Woronowicz's results concerning Peter-Weyl theory for compact quantum groups. The main new phenomenon is that for general compact quantum groups (more precisely, those which are not of Kac type), not all irreducible projective representations have to be finite-dimensional. As applications, we consider the theory of projective representations for the compact quantum groups associated with group von Neumann algebras of discrete groups, and consider a certain non-trivial projective representation for quantum SU(2).It is well known that for compact groups, one can easily extend the main theorems of the Peter-Weyl theory to cover also projective representations. In this article, we will see that if one tries to do the same for Woronowicz's compact quantum groups, one confronts at least one surprising novelty: not all irreducible projective representations of a compact quantum group have to be finite-dimensional. On the other hand, one will still be able to decompose any projective rep-

Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map

“…Namely, given a unitary 2-cocycle, we construct the associated projective regular representation containing all irreducible Ωtwisted representations and reaching thus a twisted version of the Peter-Weyl theorem. The content of this section concerns a particular case of the more general framework developed in [14] by the first author, but we give more attention here to the associated C ˚-algebraic theory.…”

“…Following [14] we have a twisted version of the Schur's orthogonality relations. This theorem follows straightforwardly by applying Lemma 3.2.6.i) with respect to rank one operators.…”

“…A projective representation theory for compact quantum groups was introduced already in [14] by the first author in terms of the so called Galois co-objects for G, which are regarded as generalized 2-cocycle functions on G. These are in bijective correspondence with the obstructions for actions G δ ñ BpHq to be inner. Particular instances of Galois co-objects are those defined in terms of (measurable) 2-cocycles Ω on G, extending the classical setting described above.…”

Product type actions ofGq

“…Set L ∞ (G q,Ω ) := L ∞ (G q ). Readers are referred to [10] for a general treatment of projective representations.…”

Quantum permutations, Hadamard matrices, and the search for matrix models

“…There are as well a few other algebraic questions, coming from the work of Collins-Härtel-Thom [43], De Commer [48], So ltan [74], Vergnioux [80] and Voigt [82].…”