Cartan subgroup
of a group 
A maximal nilpotent subgroup 
of 
each normal subgroup of finite index of which has finite index in its normalizer in 
. If 
is a connected linear algebraic group over a field of characteristic zero, then a Cartan subgroup of 
can also be defined as a closed connected subgroup whose Lie algebra is a Cartan subalgebra of the Lie algebra of 
. An example of a Cartan subgroup is the subgroup 
of all diagonal matrices in the group 
of all non-singular matrices.
In a connected linear algebraic group 
, a Cartan subgroup can also be defined as the centralizer of a maximal torus of 
, or as a connected closed nilpotent subgroup 
which coincides with the connected component of the identity (the identity component) of its normalizer in 
. The sets 
and 
of all semi-simple and unipotent elements of 
(see Jordan decomposition) are closed subgroups in 
, and 
. In addition, 
is the unique maximal torus of 
lying in 
. The dimension of a Cartan subgroup of 
is called the rank of 
. The union of all Cartan subgroups of 
contains an open subset of 
with respect to the Zariski topology (but is not, in general, the whole of 
). Every semi-simple element of 
lies in at least one Cartan subgroup, and every regular element in precisely one Cartan subgroup. If 
is a surjective morphism of linear algebraic groups, then the Cartan subgroups of 
are images with respect to 
of Cartan subgroups of 
. Any two Cartan subgroups of 
are conjugate. A Cartan subgroup of a connected semi-simple (or, more generally, reductive) group 
is a maximal torus in 
.
Let the group 
be defined over a field 
. Then there exists in 
a Cartan subgroup which is also defined over 
; in fact, 
is generated by its Cartan subgroups defined over 
. Two Cartan subgroups of 
defined over 
need not be conjugate over 
(but in the case when 
is a solvable group, they are conjugate). The variety of Cartan subgroups of 
is rational over 
.
Let 
be a connected real Lie group with Lie algebra 
. Then the Cartan subgroups of 
are closed in 
(but not necessarily connected) and their Lie algebras are Cartan subalgebras of 
. If 
is an analytic subgroup in 
and 
is the smallest algebraic subgroup of 
containing 
, then the Cartan subgroups of 
are intersections of 
with the Cartan subgroups of 
. In the case when 
is compact, the Cartan subgroups are connected, Abelian (being maximal tori) and conjugate to one another, and every element of 
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 |
Cartan subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartan_subgroup&oldid=11216