Homogeneous space

Homogeneous space - many $M$ ${\ displaystyle M}$ $M$ together with the transitive action of some group given on it $G$ ${\ displaystyle G}$ $G$ . Elements of the set M are called points of a homogeneous space, the group $G$ ${\ displaystyle G}$ $G$ - a group of movements , or the main group of a homogeneous space.

Content

1 Related Definitions
2 Examples
3 Properties
4 Literature

Related Definitions

Any point $x$ ${\ displaystyle x}$ homogeneous space $M$ ${\ displaystyle M}$ defines a subgroup
$G_{x}=\{g\in G|gx=x\}$ ${\ displaystyle G_ {x} = \ {g \ in G | gx = x \}}$

main group

G

{\ displaystyle G}

. It is called an isotropy group , or a stationary subgroup , or a point stabilizer

x

{\ displaystyle x}

. Stabilizers of different points are paired in a group

G

{\ displaystyle G}

using internal automorphisms .

Examples

With an arbitrary subgroup $H$ ${\ displaystyle H}$ groups $G$ ${\ displaystyle G}$ connected some homogeneous group space $G$ ${\ displaystyle G}$ - a bunch of $M=G/H$ ${\ displaystyle M = G / H}$ left group adjacency classes $G$ ${\ displaystyle G}$ by subgroup $H$ ${\ displaystyle H}$ , on which $G$ ${\ displaystyle G}$ acts according to the formula
$g(aH)=(ga)H$ ${\ displaystyle g (aH) = (ga) H}$ , $g,a\in G$ ${\ displaystyle g, a \ in G}$ .

This homogeneous space is called the quotient space of the group

G

{\ displaystyle G}

by subgroup

H

{\ displaystyle H}

, and the subgroup

H

{\ displaystyle H}

turns out to be a point stabilizer

eH=H

{\ displaystyle eH = H}

this space (

e

{\ displaystyle e}

- unit of group

G

{\ displaystyle G}

)

Properties

Any homogeneous space $M$ ${\ displaystyle M}$ groups $G$ ${\ displaystyle G}$ can be identified with the factor space of the group $G$ ${\ displaystyle G}$ by subgroup $H=G_{x}$ ${\ displaystyle H = G_ {x}}$ fixed point stabilizer $x\in M$ ${\ displaystyle x \ in M}$ .

If the group $G$ {\ displaystyle G} is a topological group , and $H$ {\ displaystyle H} - its subgroup (in particular, if $G$ {\ displaystyle G} - Lee's group , and $H$ {\ displaystyle H} Is a closed subgroup of $G$ {\ displaystyle G} ), then the factor space $M$ $=$ $G$ $/$ $H$ {\ displaystyle M = G / H} canonically equipped with the structure of a topological space (respectively, the structure of an analytic manifold), with respect to which the action of $G$ {\ displaystyle G} on $M$ {\ displaystyle M} is continuous (respectively analytical).
- If the group is Lee $G$ ${\ displaystyle G}$ transitively and analytically acts on an analytic manifold $M$ ${\ displaystyle M}$ then for any point $x\in M$ ${\ displaystyle x \ in M}$ subgroup $H=G_{x}$ ${\ displaystyle H = G_ {x}}$ closed and the above bijection is analytic; if at the same time the number of connected components of the group $G$ ${\ displaystyle G}$ no more than countable , then this bijection is a diffeomorphism .

Literature

Balashchenko V.V., Nikonorov Yu.G., Rodionov E.D., Slavsky V.V. Homogeneous spaces: theory and applications. - 2008.