nLab
homogeneous space

Contents

Contents

Definition

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

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

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

Examples

Properties

Relation to coset spaces

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

References

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

Last revised on April 8, 2021 at 01:56:40. See the history of this page for a list of all contributions to it.