View source for Homogeneous space of an algebraic group

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An [[Algebraic variety|algebraic variety]] $  M $ 
together with a regular transitive action of an [[Algebraic group|algebraic group]] $  G $ 
given on it. If $  x \in M $ , 
then the [[Isotropy group|isotropy group]] $  G _{x} $ 
is closed in $  G $ . 
Conversely, if $  H $ 
is a closed subgroup of an algebraic group $  G $ , 
then the set of left cosets $  G /H $ 
has the structure of an algebraic variety, making it into a homogeneous space of the algebraic group $  G $ , 
where the natural mapping $  \pi : \  G \rightarrow G / H $ 
is regular, separable and has the following universal property: For any morphism $  \phi : \  G \rightarrow X $ 
constant on cosets, there is a morphism $  \psi : \  G / H \rightarrow X $ 
such that $  \psi \pi = \phi $ . 
If $  M $ 
is any homogeneous space of the algebraic group $  G $ 
and $  H = G _{x} $ 
for some $  x \in M $ , 
then the natural bijection $  \psi : \  G / H \rightarrow M $ 
is regular, and if the ground field $  K $ 
has characteristic 0, then $  \psi $ 
is biregular (see [[#References|[1]]], [[#References|[3]]]).

Suppose that the connected group $  G $ , 
the homogeneous space $  M $ 
and the action of $  G $ 
on $  M $ 
are defined over some subfield $  k \subset K $ . 
Then the group of $  k $ -
rational points $  G (k) $ 
takes $  M (k) $ 
into itself and $  G (k) _{x} = G _{x} (k) $ 
for $  x \in M (k) $ . 
If $  k $ 
is finite, then $  M (k) \neq \emptyset $ , 
and if moreover the isotropy group $  G _{x} $ 
is connected, then $  G (k) $ 
acts transitively on $  M (k) $ . 
In the general case, the study of the $  k $ -
rational points in $  M $ 
reduces to problems in the theory of [[Galois cohomology|Galois cohomology]] (see [[#References|[2]]]).

A homogeneous space of an algebraic group $  G $ 
is always a smooth quasi-projective variety (see [[#References|[5]]]). If $  G $ 
is an affine algebraic group, then the variety $  G / H $ 
is projective if and only if $  H $ 
is a [[Parabolic subgroup|parabolic subgroup]] in $  G $ (
see [[#References|[3]]]). If $  G $ 
is reductive, then the variety $  G / H $ 
is affine if and only if the subgroup $  H $ 
is reductive (see [[Matsushima criterion|Matsushima criterion]]). A description is also known of the closed subgroups $  H $ 
of a linear algebraic group $  G $ 
over an algebraically closed field of characteristic 0 for which $  G / H $ 
is quasi-affine (see [[#References|[4]]], [[#References|[6]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) {{MR|0180551}} {{ZBL|0128.26303}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.A. Sukhanov, "A description of observable subgroups of linear algebraic groups" ''Math. USSR-Sb.'' , '''68''' (Forthcoming) ''Mat. Sb.'' , '''137''' : 1 (1988) pp. 90–102</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> W. Chow, "On the projective embedding of homogeneous varieties" , ''Algebraic topology; symposium in honour of S. Lefschetz'' , Princeton Univ. Press (1957) pp. 122–128 {{MR|0084851}} {{ZBL|0091.33302}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> G.P. Hochschild, "Basic theory of algebraic groups and Lie algebras" , Springer (1981) {{MR|0620024}} {{ZBL|0589.20025}} </TD></TR></table>