Induced coactions along a homomorphism of locally compact quantum groups

TL;DR

This paper develops a generalized theory of induced coactions for locally compact quantum groups along homomorphisms that are not necessarily closed subgroups, extending previous constructions and exploring their properties.

Contribution

It introduces a new framework for induced coactions along quantum group homomorphisms satisfying a properness condition, generalizing Vaes' approach and including fixed point algebras.

Findings

Induced coactions retain properties like imprimitivity and adjointness.

Established a base change formula relating induced coactions and restriction.

Provided an example showing limitations of 1-categories of coactions in recovering quantum groups.

Abstract

We consider induced coactions on C*-algebras along a homomorphism of locally compact quantum groups which need not give a closed quantum subgroup. Our approach generalizes the induced coactions constructed by Vaes, and also includes certain fixed point algebras. We focus on the case when the homomorphism satisfies a quantum analogue of properness. Induced coactions along such a homomorphism still admit the natural formulations of various properties known in the case of a closed quantum subgroup, such as imprimitivity and adjointness with restriction. Also, we show a relationship of induced coactions and restriction which is analogous to base change formula for modules over algebras. As an application, we give an example that shows several kinds of 1-categories of coactions with forgetful functors cannot recover the original quantum group.