nLab homogeneous space

Redirected from "homogeneous spaces".
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

Textbook accounts:

On homogeneous spaces with the same rational cohomology as a product of n-spheres:

A category theoretic 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:

On coset spaces (homogeneous spaces) and their Maurer-Cartan forms in application to first-order formulation of (super-)gravity:

Last revised on June 25, 2024 at 21:10:20. See the history of this page for a list of all contributions to it.