# Journal of Operator Theory

Volume 80, Issue 2, Fall 2018 pp. 415-428.

Compact quantum groups with representations
of bounded degree

**Authors**:
Jacek Krajczok (1) and Piotr M. Soltan (2)

**Author institution:** (1) Department of Mathematical Methods in Physics,
Faculty of Physics, University of Warsaw, Poland

(2) Department of Mathematical Methods in Physics, Faculty of Physics,
University of Warsaw, Poland

**Summary: ** We show that a compact quantum group whose all
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 the spectrum of these operators and the
dimensions of some of their eigenspaces in relation to the quantum dimension
of the corresponding irreducible representation.

**DOI: **http://dx.doi.org/10.7900/jot.2017oct23.2181

**Keywords: ** compact quantum group, representation, dimension,
quantum dimension

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