Actions of compact quantum groups

TL;DR

This paper explores the theory of actions of compact quantum groups on C*-algebras, covering foundational concepts, duality, special actions, and their classifications, with applications to quantum orthogonal groups.

Contribution

It provides a comprehensive overview of actions of compact quantum groups, introduces the concept of quantum torsors, and classifies homogeneous actions for free orthogonal quantum groups.

Findings

Homogeneous actions can be extended to free actions with classical quantum orbits.

The duality between crossed and smash product C*-algebras is established.

Classification of homogeneous actions for free orthogonal quantum groups is achieved.

Abstract

These lecture notes, prepared for the summer school "Topological quantum groups", Bedlewo 2015, deal with aspects of the theory of actions of compact quantum groups on C*-algebras ('locally compact quantum spaces'). After going over the basic notions of isotypical components and reduced and universal completions, we look at crossed and smash product C*-algebras, up to the statement of the Takesaki-Takai-Baaj-Skandalis duality (in the algebraic setting). We then look at two special types of actions, namely homogeneous actions and free actions. We study the actions which combine both types, the quantum torsors, and show that more generally any homogeneous action can be completed to a free action with a discrete, classical set of `quantum orbits'. We end with a combinatorial description of the homogeneous actions for the free orthogonal quantum groups.