Actions of compact quantum groups

Abstract

These lecture notes 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. 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.

On C∗‐completions of discrete quantum group rings

“…Note that admits a maximal ‐norm. This in fact extends to any ‘algebraic core’ of an action of a compact quantum group, as is shown, for example, in [, Proposition 4.1, Theorem 4.2]. This means that Proposition applies to algebras of such type and naturally we can consider the uniqueness of their ‐completions.…”

Quantum actions on discrete quantum spaces and a generalization of Clifford’s theory of representations

“…We shall often write L ∞ (X) for M, thus defining a "quantum (measure) space" X. We will also write Pol(X) for the algebraic core of L ∞ (X) (see [DC16;Pod95]). It is a dense unital * -subalgebra of L ∞ (X) (called also the Podleś subalgebra) such that α restricts to a coaction α Pol(X) : Pol(X) → Pol(X) ⊗ alg Pol(G) of the Hopf algebra Pol(G).…”

“…For more information on actions of compact quantum groups and related topics we refer the reader to the lecture notes [DC16]. The topics of quantum subgroups are thoroughly covered in [Daw+12] and information on normal quantum subgroups, inner automorphisms etc.…”

Part III, Free Actions of Compact Quantum Groups on C^*-Algebras

“…We rely on the C * -algebraic notion of compact quantum groups as introduced by Woronowicz [34]. For an introduction and further details we recommend [6,18,29]. A compact quantum group is given by a unital C * -algebra G together with a (usually implicit) faithful, unital * -homomorphism ∆ : G → G ⊗ G satisfying the identity (∆ ⊗ id) • ∆ = (id ⊗∆) • ∆ and such that ∆(G)(1 ⊗ G) is dense in G ⊗ G. It can be shown that there is a unique state h : G → C such that (id ⊗h) • ∆ = h = (h ⊗ id) • ∆ (see [34]).…”

“…This state is called the Haar state of G. It is not faithful in general but via the GNS-construction we may replace G by its reduced version on which the Haar state is faithful. Since G and its reduce version behave identically with respect to their representation theory and their actions (see [6,Section 4]), we will throughout the text assume that the Haar state on G is faithful.…”