nLab
homogeneous space

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 between homogenous spaces and coset spaces

Under weak topological conditions (cf. Hegason), 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

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

Revised on March 29, 2016 05:20:56 by Urs Schreiber (195.37.209.180)