2 5 M ar 1 99 8 Notes on Compact Quantum Groups

Compact quantum groups have been studied by several authors and from different points of view. The difference lies mainly in the choice of the axioms. In the end, the main results turn out to be the same. Nevertheless, the starting point has a strong influence on how the main results are obtained and on showing that certain examples satisfy these axioms. In these notes, we follow the approach of Woronowicz and we treat the compact quantum groups in the C-algebra framework. This is a natural choice when compact quantum groups are seen as a special case of locally compact quantum groups. A deep understanding of compact quantum groups in this setting is very important for understanding the problems that arise in developing a theory for locally compact quantum groups. We start with a discussion on locally compact quantum groups in general but mainly to motivate the choice of the axioms for the compact quantum groups. Then we develop the theory. We give the main examples and we show how they fit into this framework. The paper is an expository paper. It does not contain many new results although some of the proofs are certainly new and more elegant than the existing ones. Moreover, we have chosen to give a rather complete and self-contained treatment so that the paper can also serve as an introductory paper for non-specialists. Different aspects can be learned from these notes and a great deal of insight can be obtained. ∗Research Assistent of the National Fund for Scientific Research (Belgium)