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=21821