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An [[Algebraic variety|algebraic variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h0477001.png" /> together with a regular transitive action of an [[Algebraic group|algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h0477002.png" /> given on it. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h0477003.png" />, then the [[Isotropy group|isotropy group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h0477004.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h0477005.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h0477006.png" /> is a closed subgroup of an algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h0477007.png" />, then the set of left cosets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h0477008.png" /> has the structure of an algebraic variety, making it into a homogeneous space of the algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h0477009.png" />, where the natural mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770010.png" /> is regular, separable and has the following universal property: For any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770011.png" /> constant on cosets, there is a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770014.png" /> is any homogeneous space of the algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770016.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770017.png" />, then the natural bijection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770018.png" /> is regular, and if the ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770019.png" /> has characteristic 0, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770020.png" /> is biregular (see [[#References|[1]]], [[#References|[3]]]).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770021.png" />, the homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770022.png" /> and the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770024.png" /> are defined over some subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770025.png" />. Then the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770026.png" />-rational points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770027.png" /> takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770028.png" /> into itself and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770029.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770030.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770031.png" /> is finite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770032.png" />, and if moreover the isotropy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770033.png" /> is connected, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770034.png" /> acts transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770035.png" />. In the general case, the study of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770036.png" />-rational points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770037.png" /> reduces to problems in the theory of [[Galois cohomology|Galois cohomology]] (see [[#References|[2]]]).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770038.png" /> is always a smooth quasi-projective variety (see [[#References|[5]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770039.png" /> is an affine algebraic group, then the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770040.png" /> is projective if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770041.png" /> is a [[Parabolic subgroup|parabolic subgroup]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770042.png" /> (see [[#References|[3]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770043.png" /> is reductive, then the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770044.png" /> is affine if and only if the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770045.png" /> is reductive (see [[Matsushima criterion|Matsushima criterion]]). A description is also known of the closed subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770046.png" /> of a linear algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770047.png" /> over an algebraically closed field of characteristic 0 for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047700/h04770048.png" /> is quasi-affine (see [[#References|[4]]], [[#References|[6]]]).
+
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====
 
====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>
 
<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>

Latest revision as of 08:31, 16 December 2019

An algebraic variety $ M $ together with a regular transitive action of an algebraic group $ G $ given on it. If $ x \in M $ , then the 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 [1], [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 (see [2]).

A homogeneous space of an algebraic group $ G $ is always a smooth quasi-projective variety (see [5]). If $ G $ is an affine algebraic group, then the variety $ G / H $ is projective if and only if $ H $ is a parabolic subgroup in $ G $ ( see [3]). If $ G $ is reductive, then the variety $ G / H $ is affine if and only if the subgroup $ H $ is reductive (see 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 [4], [6]).

References

[1] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2] J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) MR0180551 Zbl 0128.26303
[3] J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039
[4] 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
[5] W. Chow, "On the projective embedding of homogeneous varieties" , Algebraic topology; symposium in honour of S. Lefschetz , Princeton Univ. Press (1957) pp. 122–128 MR0084851 Zbl 0091.33302
[6] G.P. Hochschild, "Basic theory of algebraic groups and Lie algebras" , Springer (1981) MR0620024 Zbl 0589.20025
How to Cite This Entry:
Homogeneous space of an algebraic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homogeneous_space_of_an_algebraic_group&oldid=44250
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article