Equivariant covering spaces of quantum homogeneous spaces

TL;DR

This paper develops a theory of equivariant extensions of C*-algebras for quantum groups, focusing on quantum homogeneous spaces, and provides classification results and index formulas for these structures.

Contribution

It introduces a fundamental framework for equivariant finite extensions of C*-algebras in the quantum setting, including a Tannaka-Krein type theorem and classification of quantum subgroups.

Findings

Every Jones' value appears as an index of an equivariant conditional expectation.

Provides an imprimitivity theorem for certain quantum groups and deformations.

Classifies finite index discrete quantum subgroups of dG_q.

Abstract

We develop a fundamental theory of compact quantum group equivariant finite extensions of C*-algebras. In particular we focus on the case of quantum homogeneous spaces and give a Tannaka-Krein type result for equivariant correspondences. As its application, we show that every Jones' value appears as the index of an equivariant conditional expectation. In the latter half of this paper, we give an imprimitivity theorem in some cases: for general compact quantum groups under a finiteness conditions, and for the Drinfeld-Jimbo deformation $G_q$ of a simply-connected compact Lie group $G$. As an application, we give a complete classification of finite index discrete quantum subgroups of $\widehat{G_q}$.