Embeddable Quantum Homogeneous Spaces

TL;DR

This paper explores the concept of quantum homogeneous spaces within locally compact quantum groups, proposing a new definition for embeddable quantum homogeneous spaces and examining their properties and examples.

Contribution

It introduces a novel definition of embeddable quantum homogeneous spaces, extending classical notions to the quantum setting and analyzing their duality and examples.

Findings

Duality for quantum homogeneous spaces on von Neumann algebras

Definition of embeddable quantum homogeneous spaces

Examples include quantum quotients and trivial bundles

Abstract

We discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups. On the von Neumann algebra level we find an interesting duality for such objects. A definition of a quantum homogeneous space is proposed along the lines of the pioneering work of Vaes on induction and imprimitivity for locally compact quantum groups. The concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces. Among various examples we single out the quantum analog of the quotient of the Cartesian product of a quantum group with itself by the diagonal subgroup, analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles.