Compact quantum groups with representations of bounded degree

TL;DR

This paper proves that compact quantum groups with uniformly bounded irreducible representation dimensions are of Kac type, meaning their Haar measure is a trace, by analyzing operators related to the Haar measure's modular properties.

Contribution

It establishes a new characterization of compact quantum groups of Kac type based on the boundedness of irreducible representation dimensions.

Findings

Such quantum groups are necessarily of Kac type.

The proof involves spectral analysis of operators linked to the Haar measure.

The dimension of eigenspaces relates to the quantum dimension of representations.

Abstract

We show that a compact quantum group all whose irreducible representations have dimension bounded by a fixed constant must be of Kac type, in other words, its Haar measure is a trace. The proof is based on establishing several facts concerning operators related to modular properties of the Haar measure. In particular we study spectrum of these operators and the dimension of some of their eigenspaces in relation to the quantum dimension of the corresponding irreducible representation.