On projective representations for compact quantum groups

TL;DR

This paper explores projective representations of compact quantum groups, extending Woronowicz's Peter-Weyl theory, and reveals new phenomena such as infinite-dimensional irreducible representations for non-Kac type quantum groups.

Contribution

It generalizes Peter-Weyl theory to projective representations of compact quantum groups, highlighting differences for non-Kac type groups and providing specific examples.

Findings

Not all irreducible projective representations are finite-dimensional for non-Kac type quantum groups.

Extended Peter-Weyl theory applies to projective representations of compact quantum groups.

Constructed a non-trivial projective representation for quantum SU(2).

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' 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 to group von Neumann algebras of discrete groups, and consider a certain non-trivial projective representation for quantum SU(2).