homogeneous space



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.



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).


  • 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 (