Cartan subalgebra

In general, a subalgebra is called toral if it consists of semisimple elements. Over an algebraically closed field, a toral subalgebra is automatically abelian. Thus, over an algebraically closed field of characteristic zero, a Cartan subalgebra can also be defined as a maximal toral subalgebra.

Kac–Moody algebras and generalized Kac–Moody algebras also have subalgebras that play the same role as the Cartan subalgebras of semisimple Lie algebras (over a field of characteristic zero).

Cartan subalgebras exist for finite-dimensional Lie algebras whenever the base field is infinite. One way to construct a Cartan subalgebra is by means of a regular element. Over a finite field, the question of the existence is still open.[citation needed]

In a finite-dimensional Lie algebra over an algebraically closed field of characteristic zero, all Cartan subalgebras are conjugate under automorphisms of the algebra, and in particular are all isomorphic. The common dimension of a Cartan subalgebra is then called the rank of the algebra.

(As noted earlier, a Cartan subalgebra can in fact be characterized as a subalgebra that is maximal among those having the above two properties.)

Over a non-algebraically closed field not every semisimple Lie algebra is splittable, however.

A Cartan subgroup of a Lie group is one of the subgroups whose Lie algebra is a Cartan subalgebra. The identity component of a subgroup has the same Lie algebra. There is no standard convention for which one of the subgroups with this property is called the Cartan subgroup, especially in the case of disconnected groups. A Cartan subgroup of a compact connected Lie group is a maximal connected Abelian subgroup (a maximal torus). Its Lie algebra is a Cartan subalgebra.

For disconnected compact Lie groups there are several inequivalent definitions of a Cartan subgroup. The most common seems to be the one given by David Vogan, who defines a Cartan subgroup to be the group of elements that normalize a fixed maximal torus and fix the fundamental Weyl chamber. This is sometimes called the large Cartan subgroup. There is also a small Cartan subgroup, defined to be the centralizer of a maximal torus. These Cartan subgroups need not be abelian in general.