nLab
homogeneous space

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

Context

Geometry

higher geometry / derived geometry

Ingredients

Concepts

Constructions

Examples

Theorems

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.

Examples

A special case of homogeneous spaces are coset spaces arising from the quotient G/H of a group G by a subgroup. For the case of Lie groups this is also called Klein geometry.

Properties

Under weak topological conditions (cf. Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces), every topological homogeneous space M is isomorphic to a coset space G/H for a closed subgroup H⊂G (the stabilizer of a fixed point in X).

Revised on November 11, 2012 23:40:41 by Zoran Škoda (193.55.36.32)
Edit | Back in time (4 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: Lie theory, group, group theory, Ab, abelian group, group algebra, 2-group, cat-n-group, finite group, semidirect product group, topological group, torsor, k-tuply groupal n-groupoid, group object, orbit category, torsion subgroup, Erlangen program, Lie group, compact Lie group, quasigroup, Hochschild-Serre spectral sequence, group cohomology, n-group, strict 2-group, groupoid object in an (infinity,1)-category, Nielsen-Schreier theorem, free group, flag variety, braid group, normal subgroup, normalizer, orbit, general linear group, nonabelian Hodge theory, special orthogonal group, orthogonal group, Poincare group, building, group extension, loop group, unitary group, local Lie group, subgroup, infinity-group, formal dg-algebra, coset, Poisson Lie group, Klein geometry, Monster group, sporadic finite simple group, classification of finite simple groups, simple group, Lie 2-group, E8, Lie group cohomology, symplectic group, mapping class group, Heisenberg group, exceptional Lie group, quantum homogeneous space, symmetric space, Lorentz group, projective unitary group, transitive action, circle n-group, symmetry, group theory - contents, Gruppenpest, group of units, locally compact topological group, divisible group, p-divisible group, Connections, Curvature, and Cohomology, iterated loop space object, Picard group, contents of contents, Cartan connection, G2, 3-group, spin^c, group of order 2, maximal compact subgroup, anti de Sitter group, Picard 3-group, sylleptic 3-group, symmetric 2-group, abelian infinity-group, induced character, infinity-group of units, conjugacy class, infinite-dimensional Lie group, free action, almost connected topological group, Coxeter group, E6, E7, compact topological group, identity component, k-monoidal infinity-group, abelian 3-group, spin representation, Kac-Moody group, order of a group, metaplectic group, Baer sum, maximal torus, group of Lie type, p-torsion, cyclic group, geometric group theory, free abelian group, infinity-group extension, commutator subgroup, abelianization, group commutator, bilinear map, tensor product of abelian groups, Feit-Thompson theorem, augmentation ideal, quotient group, anabelian group, Cauchy group theorem, index of a subgroup, Galois cohomology, finite abelian group, braided 2-group, braided infinity-group, braided 3-group, symmetric 3-group, smooth 2-group, Picard 2-group, Picard infinity-group, Weyl group