Closed quantum subgroups of locally compact quantum groups

TL;DR

This paper compares two definitions of closed quantum subgroups in locally compact quantum groups, establishing their equivalence in key cases and linking them to classical harmonic analysis results.

Contribution

It analyzes and relates two existing definitions of closed quantum subgroups, proving their equivalence in classical and quantum cases, and connects these concepts to the Herz restriction theorem.

Findings

Proves equivalence of two definitions in classical and quantum cases

Relates quantum subgroup definitions to classical harmonic analysis

Provides a new proof of Herz restriction theorem

Abstract

We investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group. Two definitions - one due to S.Vaes and one due to S.L.Woronowicz - are analyzed and relations between them discussed. Among many reformulations we prove that the former definition can be phrased in terms of quasi-equivalence of representations of quantum groups while the latter can be related to an old definition of Podle\'s from the theory of compact quantum groups. The cases of classical groups, duals of classical groups, compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context. A deep relationship with the quantum group generalization of Herz restriction theorem from classical harmonic analysis is also established, in particular, in the course of our analysis we give a new proof of Herz restriction theorem.