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. For a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero, there is a simpler approach: by definition, a toral subalgebra is a subalgebra of that consists of semisimple elements. A Cartan subalgebra of is then the same thing as a maximal toral subalgebra and the existence of a maximal toral subalgebra is easy to see. 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. For a finite-dimensional complex semisimple Lie algebra, the existence of a Cartan subalgebra is much simpler to establish, assuming the existence of a compact real form. In that case, may be taken as the complexification of the Lie algebra of a maximal torus of the compact group. Kac–Moody algebras and generalized Kac–Moody algebras also have Cartan subalgebras.
Cartan subalgebras of semisimple Lie algebras
A Cartan subalgebra of a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic 0 is abelian and also has the following property of its adjoint representation: the weight eigenspaces of restricted to diagonalize the representation, and the eigenspace of the zero weight vector is. The non-zero weights are called the roots, and the corresponding eigenspaces are called root spaces, and are all 1-dimensional. If is a linear Lie algebra over an algebraically closed field, then any Cartan subalgebra of is the centralizer of a maximal toral subalgebra of. If is semisimple and the field has characteristic zero, then a maximal toral subalgebra is self-normalizing, and so is equal to the associated Cartan subalgebra. If in addition is semisimple, then the adjoint representation presents as a linear Lie algebra, so that a subalgebra of is Cartan if and only if it is a maximal toral subalgebra.
Examples
Any nilpotent Lie algebra is its own Cartan subalgebra.
A Cartan subalgebra of the Lie algebra of n×n matrices over a field is the algebra of all diagonal matrices.
The Lie algebra sl2 of 2 by 2 matrices of trace 0 has two non-conjugate Cartan subalgebras.
The dimension of a Cartan subalgebra is not in general the maximal dimension of an abelian subalgebra, even for complex simple Lie algebras. For example, the Lie algebra sl2n of 2n by 2n matrices of trace 0 has a Cartan subalgebra of rank 2n−1 but has a maximal abelian subalgebra of dimension n2consisting of all matrices of the form with A any n by n matrix. One can directly see this abelian subalgebra is not a Cartan subalgebra, since it is contained in the nilpotent algebra of strictly upper triangular matrices.
Splitting Cartan subalgebra
Over non-algebraically closed fields, not all Cartan subalgebras are conjugate. An important class are splitting Cartan subalgebras: if a Lie algebra admits a splitting Cartan subalgebra then it is called splittable, and the pair is called a split Lie algebra; over an algebraically closed field every semisimple Lie algebra is splittable. Any two splitting Cartan algebras are conjugate, and they fulfill a similar function to Cartan algebras in semisimple Lie algebras over algebraically closed fields, so split semisimple Lie algebras share many properties with semisimple Lie algebras over algebraically closed fields. Over a non-algebraically closed field not every semisimple Lie algebra is splittable, however.
