# Contents

## Definition

Given a topological group or algebraic group or Lie group, etc., $G$, a homogeneous $G$-space is a topological space or scheme, or smooth manifold etc. with transitive $G$-action.

A principal homogeneous $G$-space is the total space of a $G$-torsor over a point.

There are generalizations, e.g. the quantum homogeneous space for the case of quantum groups.

## Properties

### Relation to coset spaces

Under weak topological conditions (cf. Helgason), every topological homogeneous space $M$ is isomorphic to a coset space $G/H$ for a closed subgroup $H\subset G$ (the stabilizer of a fixed point in $X$).

## References

• Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces

The following article has categorical analysis of relation between the total space of a principal bundle and of the corresponding quotient space both for the classical case and for noncommutative generalizations:

• Tomasz Brzeziński, On synthetic interpretation of quantum principal bundles, AJSE D - Mathematics 35(1D): 13-27, 2010 arxiv:0912.0213; Quantum group differentials, bundles and gauge theory, Encyclopedia of Mathematical Physics, Acad. Press. 2006, pp. 236–244 doi

Revised on May 3, 2016 09:08:56 by Zoran Škoda (161.53.28.4)