View source for Cartan subgroup

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''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|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|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====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  C. Chevalley,   "Theory of Lie groups" , '''1''' , Princeton Univ. Press  (1946)  {{MR|0082628}} {{MR|0015396}} {{ZBL|0063.00842}} </TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  C. Chevalley,   "Théorie des groupes de Lie" , '''2–3''' , Hermann  (1951–1955)  {{MR|0068552}} {{MR|0051242}} {{MR|0019623}} {{ZBL|0186.33104}} {{ZBL|0054.01303}} {{ZBL|0063.00843}} </TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  A. Borel,   J. Tits,   "Groupes réductifs"  ''Publ. Math. IHES'' , '''27'''  (1965)  pp. 55–150  {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Demazure,   A. Grothendieck,   "Schémas en groupes I-III" , ''Lect. notes in math.'' , '''151–153''' , Springer  (1970)    {{MR|}} {{ZBL|}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Borel,   T.A. Springer,   "Rationality properties of linear algebraic groups"  ''Tohoku Math. J. (2)'' , '''20'''  (1968)  pp. 443–497  {{MR|0244259}} {{ZBL|0211.53302}} </TD></TR></table>