nLab
orbit

Contents

Contents

Definition

Discrete case

Given an action G×XXG\times X\to X of a (discrete) group GG on a set XX, any set of the form Gx={gx|gG}G x = \{g x|g\in G\} for a fixed xXx\in X is called an orbit of the action, or the GG-orbit through the point xx. The set XX is a disjoint union of its orbits.

Category of orbits

The category of orbits of a group GG is the full subcategory of the category of sets with an action of GG.

Since any orbit of GG is isomorphic to the orbit G/HG/H for some group HH, the category of GG-orbits admits the following alternative description: its objects are subgroups HH of GG and morphisms H 1H 2H_1\to H_2 are elements [g]G/H 2[g]\in G/H_2 such that H 1gH 2g 1H_1\subset gH_2g^{-1}.

In particular, the group of automorphisms of a GG-orbit G/HG/H is N G(H)/HN_G(H)/H, where N G(H)N_G(H) is the normalizer of HH in GG.

Topological case

If GG is a topological group, XX a topological space and the action continuous, then one can distinguish closed orbits from those which are not. Even when one starts with G,XG,X Hausdorff, the space of orbits is typically non-Hausdorff. (This problem is one of the motivations of the noncommutative geometry of Connes’ school.)

If the original space is paracompact Hausdorff, then every orbit GxG x as a topological GG-space is isomorphic to G/HG/H, where HH is the stabilizer subgroup of xx.

Orbit method

The orbit method is a method in representation theory introduced by Kirillov, Kostant and Souriau; it is a special case of geometric quantization. The orbit method is based on the study of the representations constructed from the coadjoint orbits with Kirillov symplectic structure. The terminology ‘geometric quantization’ allows for more general underlying spaces.

Given a compact Lie group KK with complexification GG and a unitary representation ρ\rho of KK on a finite-dimensional complex space VV, the real orbits of the highest weight vector agrees with the complex orbits, i.e. the orbits of the extension of this representation to the representation of the complexification. These are the coherent state orbits; there is also an infinite-dimensional version for reductive groups and representations which allows them (so-called coherent state representations).

Last revised on June 16, 2019 at 09:48:54. See the history of this page for a list of all contributions to it.