# Cartan subgroup

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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.

How to Cite This Entry:
Cartan subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartan_subgroup&oldid=11216
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article