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''of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c0205601.png" />''
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{{TEX|done}}
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''of a group $  G $ ''
  
A maximal nilpotent subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c0205602.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c0205603.png" /> each normal subgroup of finite index of which has finite index in its normalizer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c0205604.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c0205605.png" /> is a connected linear algebraic group over a field of characteristic zero, then a Cartan subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c0205606.png" /> can also be defined as a closed connected subgroup whose Lie algebra is a [[Cartan subalgebra|Cartan subalgebra]] of the Lie algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c0205607.png" />. An example of a Cartan subgroup is the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c0205608.png" /> of all diagonal matrices in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c0205609.png" /> of all non-singular matrices.
 
  
In a connected linear algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056010.png" />, a Cartan subgroup can also be defined as the centralizer of a maximal torus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056011.png" />, or as a connected closed nilpotent subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056012.png" /> which coincides with the connected component of the identity (the identity component) of its normalizer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056013.png" />. The sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056015.png" /> of all semi-simple and unipotent elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056016.png" /> (see [[Jordan decomposition|Jordan decomposition]]) are closed subgroups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056017.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056018.png" />. In addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056019.png" /> is the unique maximal torus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056020.png" /> lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056021.png" />. The dimension of a Cartan subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056022.png" /> is called the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056023.png" />. The union of all Cartan subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056024.png" /> contains an open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056025.png" /> with respect to the Zariski topology (but is not, in general, the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056026.png" />). Every semi-simple element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056027.png" /> lies in at least one Cartan subgroup, and every regular element in precisely one Cartan subgroup. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056028.png" /> is a surjective morphism of linear algebraic groups, then the Cartan subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056029.png" /> are images with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056030.png" /> of Cartan subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056031.png" />. Any two Cartan subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056032.png" /> are conjugate. A Cartan subgroup of a connected semi-simple (or, more generally, reductive) group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056033.png" /> is a maximal torus in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056034.png" />.
+
A maximal nilpotent subgroup $  C $
 +
of $  G $
 +
each normal subgroup of finite index of which has finite index in its normalizer in $  G $ .
 +
If $  G $
 +
is a connected linear algebraic group over a field of characteristic zero, then a Cartan subgroup of $  G $
 +
can also be defined as a closed connected subgroup whose Lie algebra is a [[Cartan subalgebra|Cartan subalgebra]] of the Lie algebra of $  G $ .  
 +
An example of a Cartan subgroup is the subgroup $  D $
 +
of all diagonal matrices in the group $  \mathop{\rm GL}\nolimits _{n} (k) $
 +
of all non-singular matrices.
  
Let the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056035.png" /> be defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056036.png" />. Then there exists in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056037.png" /> a Cartan subgroup which is also defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056038.png" />; in fact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056039.png" /> is generated by its Cartan subgroups defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056040.png" />. Two Cartan subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056041.png" /> defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056042.png" /> need not be conjugate over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056043.png" /> (but in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056044.png" /> is a solvable group, they are conjugate). The variety of Cartan subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056045.png" /> is rational over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056046.png" />.
+
In a connected linear algebraic group $  G $ ,
 +
a Cartan subgroup can also be defined as the centralizer of a maximal torus of $  G $ ,
 +
or as a connected closed nilpotent subgroup $  C $
 +
which coincides with the connected component of the identity (the identity component) of its normalizer in $  G $ .  
 +
The sets $  C _{s} $
 +
and $  C _{u} $
 +
of all semi-simple and unipotent elements of $  C $ (
 +
see [[Jordan decomposition|Jordan decomposition]]) are closed subgroups in $  C $ ,  
 +
and $  C = C _{s} \times C _{u} $ .  
 +
In addition, $  C _{s} $
 +
is the unique maximal torus of $  G $
 +
lying in $  C $ .  
 +
The dimension of a Cartan subgroup of $  G $
 +
is called the rank of $  G $ .  
 +
The union of all Cartan subgroups of $  G $
 +
contains an open subset of $  G $
 +
with respect to the Zariski topology (but is not, in general, the whole of $  G $ ).  
 +
Every semi-simple element of $  G $
 +
lies in at least one Cartan subgroup, and every regular element in precisely one Cartan subgroup. If $  \phi : \  G \rightarrow G ^ \prime  $
 +
is a surjective morphism of linear algebraic groups, then the Cartan subgroups of $  G ^ \prime  $
 +
are images with respect to $  \phi $
 +
of Cartan subgroups of $  G $ .  
 +
Any two Cartan subgroups of $  G $
 +
are conjugate. A Cartan subgroup of a connected semi-simple (or, more generally, reductive) group $  G $
 +
is a maximal torus in $  G $ .
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056047.png" /> be a connected real Lie group with Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056048.png" />. Then the Cartan subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056049.png" /> are closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056050.png" /> (but not necessarily connected) and their Lie algebras are Cartan subalgebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056051.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056052.png" /> is an analytic subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056054.png" /> is the smallest algebraic subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056055.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056056.png" />, then the Cartan subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056057.png" /> are intersections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056058.png" /> with the Cartan subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056059.png" />. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056060.png" /> is compact, the Cartan subgroups are connected, Abelian (being maximal tori) and conjugate to one another, and every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020560/c02056061.png" /> lies in some Cartan subgroup.
+
 
 +
Let the group $  G $
 +
be defined over a field $  k $ .
 +
Then there exists in $  G $
 +
a Cartan subgroup which is also defined over $  k $ ;
 +
in fact, $  G $
 +
is generated by its Cartan subgroups defined over $  k $ .  
 +
Two Cartan subgroups of $  G $
 +
defined over $  k $
 +
need not be conjugate over $  k $ (
 +
but in the case when $  G $
 +
is a solvable group, they are conjugate). The variety of Cartan subgroups of $  G $
 +
is rational over $  k $ .
 +
 
 +
 
 +
Let $  G $
 +
be a connected real Lie group with Lie algebra $  \mathfrak g $ .  
 +
Then the Cartan subgroups of $  G $
 +
are closed in $  G $ (
 +
but not necessarily connected) and their Lie algebras are Cartan subalgebras of $  \mathfrak g $ .  
 +
If $  G $
 +
is an analytic subgroup in $  \mathop{\rm GL}\nolimits _{n} ( \mathbf R ) $
 +
and $  \overline{G}  $
 +
is the smallest algebraic subgroup of $  \mathop{\rm GL}\nolimits _{n} ( \mathbf R ) $
 +
containing $  G $ ,  
 +
then the Cartan subgroups of $  G $
 +
are intersections of $  G $
 +
with the Cartan subgroups of $  \overline{G}  $ .  
 +
In the case when $  G $
 +
is compact, the Cartan subgroups are connected, Abelian (being maximal tori) and conjugate to one another, and every element of $  G $
 +
lies in some Cartan subgroup.
  
 
====References====
 
====References====

Latest revision as of 10:47, 17 December 2019

of a group $ G $


A maximal nilpotent subgroup $ C $ of $ G $ each normal subgroup of finite index of which has finite index in its normalizer in $ G $ . If $ G $ is a connected linear algebraic group over a field of characteristic zero, then a Cartan subgroup of $ G $ can also be defined as a closed connected subgroup whose Lie algebra is a Cartan subalgebra of the Lie algebra of $ G $ . An example of a Cartan subgroup is the subgroup $ D $ of all diagonal matrices in the group $ \mathop{\rm GL}\nolimits _{n} (k) $ of all non-singular matrices.

In a connected linear algebraic group $ G $ , a Cartan subgroup can also be defined as the centralizer of a maximal torus of $ G $ , or as a connected closed nilpotent subgroup $ C $ which coincides with the connected component of the identity (the identity component) of its normalizer in $ G $ . The sets $ C _{s} $ and $ C _{u} $ of all semi-simple and unipotent elements of $ C $ ( see Jordan decomposition) are closed subgroups in $ C $ , and $ C = C _{s} \times C _{u} $ . In addition, $ C _{s} $ is the unique maximal torus of $ G $ lying in $ C $ . The dimension of a Cartan subgroup of $ G $ is called the rank of $ G $ . The union of all Cartan subgroups of $ G $ contains an open subset of $ G $ with respect to the Zariski topology (but is not, in general, the whole of $ G $ ). Every semi-simple element of $ G $ lies in at least one Cartan subgroup, and every regular element in precisely one Cartan subgroup. If $ \phi : \ G \rightarrow G ^ \prime $ is a surjective morphism of linear algebraic groups, then the Cartan subgroups of $ G ^ \prime $ are images with respect to $ \phi $ of Cartan subgroups of $ G $ . Any two Cartan subgroups of $ G $ are conjugate. A Cartan subgroup of a connected semi-simple (or, more generally, reductive) group $ G $ is a maximal torus in $ G $ .


Let the group $ G $ be defined over a field $ k $ . Then there exists in $ G $ a Cartan subgroup which is also defined over $ k $ ; in fact, $ G $ is generated by its Cartan subgroups defined over $ k $ . Two Cartan subgroups of $ G $ defined over $ k $ need not be conjugate over $ k $ ( but in the case when $ G $ is a solvable group, they are conjugate). The variety of Cartan subgroups of $ G $ is rational over $ k $ .


Let $ G $ be a connected real Lie group with Lie algebra $ \mathfrak g $ . Then the Cartan subgroups of $ G $ are closed in $ G $ ( but not necessarily connected) and their Lie algebras are Cartan subalgebras of $ \mathfrak g $ . If $ G $ is an analytic subgroup in $ \mathop{\rm GL}\nolimits _{n} ( \mathbf R ) $ and $ \overline{G} $ is the smallest algebraic subgroup of $ \mathop{\rm GL}\nolimits _{n} ( \mathbf R ) $ containing $ G $ , then the Cartan subgroups of $ G $ are intersections of $ G $ with the Cartan subgroups of $ \overline{G} $ . In the case when $ G $ is compact, the Cartan subgroups are connected, Abelian (being maximal tori) and conjugate to one another, and every element of $ G $ lies in some Cartan subgroup.

References

[1a] C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946) MR0082628 MR0015396 Zbl 0063.00842
[1b] C. Chevalley, "Théorie des groupes de Lie" , 2–3 , Hermann (1951–1955) MR0068552 MR0051242 MR0019623 Zbl 0186.33104 Zbl 0054.01303 Zbl 0063.00843
[2] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[3] A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[4] M. Demazure, A. Grothendieck, "Schémas en groupes I-III" , Lect. notes in math. , 151–153 , Springer (1970)


Comments

References

[a1] A. Borel, T.A. Springer, "Rationality properties of linear algebraic groups" Tohoku Math. J. (2) , 20 (1968) pp. 443–497 MR0244259 Zbl 0211.53302
How to Cite This Entry:
Cartan subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartan_subgroup&oldid=21855
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article