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)

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