Open quantum subgroups of locally compact quantum groups

TL;DR

This paper introduces the concept of open quantum subgroups within locally compact quantum groups, providing multiple characterizations and exploring their properties and classifications, including their relation to dual quantum groups.

Contribution

It defines open quantum subgroups and establishes their equivalences and properties, including their correspondence with normal compact quantum subgroups of the dual.

Findings

Open quantum subgroups are characterized by group-like projections and algebra inclusions.

Normal open quantum subgroups correspond bijectively to normal compact quantum subgroups of the dual.

Open quantum subgroups are shown to be closed in the sense of Vaes.

Abstract

The notion of an open quantum subgroup of a locally compact quantum group is introduced and given several equivalent characterizations in terms of group-like projections, inclusions of quantum group C*-algebras and properties of respective quantum homogenous spaces. Open quantum subgroups are shown to be closed in the sense of Vaes and normal open quantum subgroups are proved to be in 1-1 correspondence with normal compact quantum subgroups of the dual quantum group.