Free actions of compact quantum group on unital C*-algebras

TL;DR

This paper characterizes free actions of compact quantum groups on unital C*-algebras through an algebraic isomorphism condition, extending classical Galois theory concepts to the quantum setting.

Contribution

It establishes an algebraic criterion for freeness of compact quantum group actions on C*-algebras, generalizing classical Galois and covering space results.

Findings

Freeness of quantum group actions is equivalent to a canonical map being an isomorphism.

Provides an algebraic characterization of free actions in topological and quantum contexts.

Extends classical Galois theory concepts to the realm of quantum symmetries.

Abstract

Let F be a field, G a finite group, and Map(G,F) the Hopf algebra of all set-theoretic maps G->F. If E is a finite field extension of F and G is its Galois group, the extension is Galois if and only if the canonical map resulting from viewing E as a Map(G,F)-comodule is an isomorphism. Similarly, a finite covering space is regular if and only if the analogous canonical map is an isomorphism. In this paper we extend this point of view to actions of compact quantum groups on unital C*-algebras. We prove that such an action is free if and only if the canonical map (obtained using the underlying Hopf algebra of the compact quantum group) is an isomorphism. In particular, we are able to express the freeness of a compact Hausdorff topological group action on a compact Hausdorff topological space in algebraic terms.