# 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

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