Introduction to compact (matrix) quantum groups and Banica-Speicher (easy) quantum groups

This is a transcript of a series of eight lectures, 90 min each, held at IMSc Chennai, India from 5-24 January 2015. We give basic definitions, properties and examples of compact quantum groups and compact matrix quantum groups such as the existence of a Haar state, the representation theory and Woronowicz's quantum version of the Tannaka-Krein theorem. Building on this, we define Banica-Speicher quantum groups (also called easy quantum groups), a class of compact matrix quantum groups determined by the combinatorics of set partitions. We sketch the classification of Banica-Speicher quantum groups and we list some applications. We review the state-of-the-art regarding Banica-Speicher quantum groups and we list some open problems.