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

TL;DR

This paper extends classical Clifford theory to quantum groups acting on discrete quantum spaces, establishing finite orbit relations and linking them to quantum subgroup representations.

Contribution

It generalizes Clifford's classical representation restrictions to the setting of quantum subgroups within discrete quantum groups.

Findings

Finite orbits for actions on finite-dimensional factors.

Generalization of Clifford's theory to quantum subgroups.

Connection between orbit relations and Vergnioux's equivalence relation.

Abstract

To any action of a compact quantum group on a von Neumann algebra which is a direct sum of factors we associate an equivalence relation corresponding to the partition of a space into orbits of the action. We show that in case all factors are finite-dimensional (i.e. when the action is on a discrete quantum space) the relation has finite orbits. We then apply this to generalize the classical theory of Clifford, concerning the restrictions of representations to normal subgroups, to the framework of quantum subgroups of discrete quantum groups, itself extending the context of closed normal quantum subgroups of compact quantum groups. Finally, a link is made between our equivalence relation in question and another equivalence relation defined by R.Vergnioux.