Notes on Compact Quantum Groups

TL;DR

This paper provides an accessible, detailed overview of compact quantum groups, focusing on Woronowicz's approach, and compares it with other treatments to facilitate understanding of their core concepts and examples.

Contribution

It offers a comprehensive, easy-to-understand set of notes on compact quantum groups, emphasizing Woronowicz's axiomatic approach and including detailed examples and comparisons.

Findings

Clarifies the axiomatic foundations of compact quantum groups

Provides detailed examples illustrating the theory

Compares different approaches to the subject

Abstract

We have written down a set of notes on compact quantum groups from which all the different aspects can be learned in an easy way and such that a lot of insight can be obtained without too much effort. Compact quantum groups have been studied by several authors, from different points of view. The difference lies mainly in the choice of the axioms and consequently, in the way the main results are proven. These results however are essentially the same in all these cases. In these notes, we mainly follow the approach of Woronowicz and we extensively motivate this choice. We give a complete and rather detailed treatment, starting from a simple set of axioms and obtaining the main results. We also discuss the most common examples and show how they fit into the framework. During this process, we compare with the existing other treatments.