On a certain approach to quantum homogeneous spaces

TL;DR

This paper introduces a new framework for defining quantum homogeneous spaces that generalizes classical notions and encompasses various quantum cases, including non-quotient spaces and deformations, while excluding paradoxical examples.

Contribution

It provides a comprehensive and unifying definition of quantum homogeneous spaces that extends existing concepts and is compatible with multiple quantum group constructions.

Findings

Reduces to classical homogeneous spaces in the classical limit

Includes Vaes' quotient and Podles' non-quotient spaces

Rules out paradoxical non-compact quantum spaces

Abstract

We propose a definition of a quantum homogeneous space of a locally compact quantum group. We show that classically it reduces to the notion of a homogeneous spaces. On the quantum level our definition goes beyond the quotient case. It provides a framework which, besides the Vaes' quotient of a locally compact quantum group by its closed quantum subgroup (our main motivation) is also compatible with, generically non-quotient, quantum homogeneous spaces of a compact quantum group studied by P. Podles as well as the Rieffel deformation of G-homogeneous spaces. Finally, our definition rules out the paradoxical examples of the non-compact quantum homogeneous spaces of a compact quantum group.