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An affine algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d0315501.png" /> that is isomorphic to a closed subgroup of an [[Algebraic torus|algebraic torus]]. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d0315502.png" /> is isomorphic to a closed subgroup of a multiplicative group of all diagonal matrices of given size. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d0315503.png" /> is defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d0315504.png" /> and the isomorphism is defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d0315505.png" />, the diagonalizable algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d0315506.png" /> is said to be split (or decomposable) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d0315507.png" />.
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{{TEX|done}}
 +
An affine algebraic group $  G $
 +
that is isomorphic to a closed subgroup of an [[Algebraic torus|algebraic torus]]. Thus, $  G $
 +
is isomorphic to a closed subgroup of a multiplicative group of all diagonal matrices of given size. If $  G $
 +
is defined over a field $  k $
 +
and the isomorphism is defined over $  k $ ,  
 +
the diagonalizable algebraic group $  G $
 +
is said to be split (or decomposable) over $  k $ .
  
Any closed subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d0315508.png" /> in a diagonalizable algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d0315509.png" />, as well as the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155010.png" /> under an arbitrary rational homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155011.png" />, is a diagonalizable algebraic group. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155012.png" /> is defined and split over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155013.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155014.png" /> is defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155015.png" />, then both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155017.png" /> are defined and split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155018.png" />.
 
  
A diagonalizable algebraic group is split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155019.png" /> if and only if elements in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155020.png" /> of its rational characters are rational over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155021.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155022.png" /> contains no non-unit elements rational over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155023.png" />, the diagonalizable algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155024.png" /> is said to be anisotropic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155025.png" />. Any diagonalizable algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155026.png" /> defined over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155027.png" /> is split over some finite separable extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155028.png" />.
+
Any closed subgroup  $  H $
 +
in a diagonalizable algebraic group $  G $ ,
 +
as well as the image of $  G $
 +
under an arbitrary rational homomorphism  $  \phi $ ,  
 +
is a diagonalizable algebraic group. If, in addition,  $  G $
 +
is defined and split over a field $  k $ ,
 +
while  $  \phi $
 +
is defined over  $  k $ ,
 +
then both  $  H $
 +
and  $  \phi (G) $
 +
are defined and split over $  k $ .
  
A diagonalizable algebraic group is connected if and only if it is an algebraic torus. The connectedness of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155029.png" /> is also equivalent to the absence of torsion in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155030.png" />. For any diagonalizable algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155031.png" /> defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155032.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155033.png" /> is a finitely-generated Abelian group without <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155034.png" />-torsion, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155035.png" /> is the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155036.png" />.
 
  
Any diagonalizable algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155037.png" /> which is defined and split over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155038.png" /> is the direct product of a finite Abelian group and an algebraic torus defined and split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155039.png" />. Any diagonalizable algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155040.png" /> which is connected and defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155041.png" /> contains a largest anisotropic subtorus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155042.png" /> and a largest subtorus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155043.png" /> which is split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155044.png" />; for these, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155045.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155046.png" /> is a finite set.
+
A diagonalizable algebraic group is split over  $  k $
 +
if and only if elements in the group  $  \widehat{G}  $
 +
of its rational characters are rational over $  k $ .  
 +
If  $  \widehat{G}  $
 +
contains no non-unit elements rational over  $  k $ ,
 +
the diagonalizable algebraic group $  G $
 +
is said to be anisotropic over $  k $ .  
 +
Any diagonalizable algebraic group $  G $
 +
defined over the field $  k $
 +
is split over some finite separable extension of  $  k $ .
  
If a diagonalizable algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155047.png" /> is defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155049.png" /> is the Galois group of the separable closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155050.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155051.png" /> is endowed with a continuous action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155052.png" />. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155053.png" /> is a rational homomorphism between diagonalizable algebraic groups, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155056.png" /> are defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155057.png" />, then the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155058.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155060.png" />-equivariant (i.e. is a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155061.png" />-modules). The resulting contravariant functor from the category of diagonalizable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155062.png" />-groups and their <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155063.png" />-morphisms into the category of finitely-generated Abelian groups without <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155064.png" />-torsion with a continuous action of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155065.png" /> and their <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031550/d03155066.png" />-equivariant homomorphisms is an equivalence of these categories.
+
 
 +
A diagonalizable algebraic group is connected if and only if it is an algebraic torus. The connectedness of  $  G $
 +
is also equivalent to the absence of torsion in  $  \widehat{G}  $ .  
 +
For any diagonalizable algebraic group  $  G $
 +
defined over  $  k $ ,
 +
the group  $  \widehat{G}  $
 +
is a finitely-generated Abelian group without  $  p $ -
 +
torsion, where  $  p $
 +
is the characteristic of  $  k $ .
 +
 
 +
 
 +
Any diagonalizable algebraic group  $  G $
 +
which is defined and split over a field  $  k $
 +
is the direct product of a finite Abelian group and an algebraic torus defined and split over  $  k $ .  
 +
Any diagonalizable algebraic group  $  G $
 +
which is connected and defined over a field $  k $
 +
contains a largest anisotropic subtorus  $  G _{a} $
 +
and a largest subtorus  $  G _{d} $
 +
which is split over  $  k $ ;
 +
for these,  $  G = G _{a} G _{d} $ ,
 +
and $  G _{a} \cap G _{d} $
 +
is a finite set.
 +
 
 +
If a diagonalizable algebraic group  $  G $
 +
is defined over a field  $  k $
 +
and  $  \Gamma $
 +
is the Galois group of the separable closure of $  k $ ,  
 +
then $  \widehat{G}  $
 +
is endowed with a continuous action of $  \Gamma $ .  
 +
If, in addition, $  \phi : \  G \rightarrow H $
 +
is a rational homomorphism between diagonalizable algebraic groups, while $  G $ ,  
 +
$  H $
 +
and $  \phi $
 +
are defined over $  k $ ,  
 +
then the homomorphism $  \widehat \phi  : \  \widehat{H}  \rightarrow \widehat{G}  $
 +
is $  \Gamma $ -
 +
equivariant (i.e. is a homomorphism of $  \Gamma $ -
 +
modules). The resulting contravariant functor from the category of diagonalizable $  \Gamma $ -
 +
groups and their $  k $ -
 +
morphisms into the category of finitely-generated Abelian groups without $  k $ -
 +
torsion with a continuous action of the group $  p $
 +
and their $  \Gamma $ -
 +
equivariant homomorphisms is an equivalence of these categories.
  
 
====References====
 
====References====

Latest revision as of 11:05, 17 December 2019

An affine algebraic group $ G $ that is isomorphic to a closed subgroup of an algebraic torus. Thus, $ G $ is isomorphic to a closed subgroup of a multiplicative group of all diagonal matrices of given size. If $ G $ is defined over a field $ k $ and the isomorphism is defined over $ k $ , the diagonalizable algebraic group $ G $ is said to be split (or decomposable) over $ k $ .


Any closed subgroup $ H $ in a diagonalizable algebraic group $ G $ , as well as the image of $ G $ under an arbitrary rational homomorphism $ \phi $ , is a diagonalizable algebraic group. If, in addition, $ G $ is defined and split over a field $ k $ , while $ \phi $ is defined over $ k $ , then both $ H $ and $ \phi (G) $ are defined and split over $ k $ .


A diagonalizable algebraic group is split over $ k $ if and only if elements in the group $ \widehat{G} $ of its rational characters are rational over $ k $ . If $ \widehat{G} $ contains no non-unit elements rational over $ k $ , the diagonalizable algebraic group $ G $ is said to be anisotropic over $ k $ . Any diagonalizable algebraic group $ G $ defined over the field $ k $ is split over some finite separable extension of $ k $ .


A diagonalizable algebraic group is connected if and only if it is an algebraic torus. The connectedness of $ G $ is also equivalent to the absence of torsion in $ \widehat{G} $ . For any diagonalizable algebraic group $ G $ defined over $ k $ , the group $ \widehat{G} $ is a finitely-generated Abelian group without $ p $ - torsion, where $ p $ is the characteristic of $ k $ .


Any diagonalizable algebraic group $ G $ which is defined and split over a field $ k $ is the direct product of a finite Abelian group and an algebraic torus defined and split over $ k $ . Any diagonalizable algebraic group $ G $ which is connected and defined over a field $ k $ contains a largest anisotropic subtorus $ G _{a} $ and a largest subtorus $ G _{d} $ which is split over $ k $ ; for these, $ G = G _{a} G _{d} $ , and $ G _{a} \cap G _{d} $ is a finite set.

If a diagonalizable algebraic group $ G $ is defined over a field $ k $ and $ \Gamma $ is the Galois group of the separable closure of $ k $ , then $ \widehat{G} $ is endowed with a continuous action of $ \Gamma $ . If, in addition, $ \phi : \ G \rightarrow H $ is a rational homomorphism between diagonalizable algebraic groups, while $ G $ , $ H $ and $ \phi $ are defined over $ k $ , then the homomorphism $ \widehat \phi : \ \widehat{H} \rightarrow \widehat{G} $ is $ \Gamma $ - equivariant (i.e. is a homomorphism of $ \Gamma $ - modules). The resulting contravariant functor from the category of diagonalizable $ \Gamma $ - groups and their $ k $ - morphisms into the category of finitely-generated Abelian groups without $ k $ - torsion with a continuous action of the group $ p $ and their $ \Gamma $ - equivariant homomorphisms is an equivalence of these categories.

References

[1] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2] T. Ono, "Arithmetic of algebraic tori" Ann. of Math. , 74 : 1 (1961) pp. 101–139 MR0124326 Zbl 0119.27801


Comments

References

[a1] J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039
How to Cite This Entry:
Diagonalizable algebraic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagonalizable_algebraic_group&oldid=21836
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article