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group theory

# Contents

## Definition

### Discrete case

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

### Category of orbits

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

Since any orbit of $G$ is isomorphic to the orbit $G/H$ for some group $H$, the category of $G$-orbits admits the following alternative description: its objects are subgroups $H$ of $G$ and morphisms $H_1\to H_2$ are elements $[g]\in G/H_2$ such that $H_1\subset g H_2g^{-1}$.

In particular, the group of automorphisms of a $G$-orbit $G/H$ is $N_G(H)/H$, where $N_G(H)$ is the normalizer of $H$ in $G$.

### Topological case

If $G$ is a topological group, $X$ a topological space and the action continuous, then one can distinguish closed orbits from those which are not. Even when one starts with $G,X$ 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 $G x$ as a topological $G$-space is isomorphic to $G/H$, where $H$ is the stabilizer subgroup of $x$.

### 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 $K$ with complexification $G$ and a unitary representation $\rho$ of $K$ on a finite-dimensional complex space $V$, 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 August 27, 2020 at 09:36:37. See the history of this page for a list of all contributions to it.