Equivariant representation theory for proper actions on discrete spaces

TL;DR

This paper develops a framework for understanding proper actions of locally compact quantum groups on discrete quantum spaces, revealing that such actions can be fully reconstructed from their equivariant representation theory, and establishing that these quantum groups are algebraic.

Contribution

It introduces a concrete unitary 2-category of Hilbert bimodules to classify proper quantum group actions on discrete spaces, enabling complete reconstruction of the quantum group and its action.

Findings

Proper actions lead to a unitary 2-category of bimodules.

Quantum groups acting properly are shown to be algebraic.

Complete reconstruction of quantum groups from their actions is achieved.

Abstract

Starting from any proper action of any locally compact quantum group on any discrete quantum space, we show that its equivariant representation theory yields a concrete unitary 2-category of finite type Hilbert bimodules over the discrete quantum space, from which the quantum group and its action may be completely reconstructed as in a previous article by the author. In particular, this shows that any locally compact quantum group acting properly on a discrete quantum space must be an algebraic quantum group.