In a finite-dimensional
semisimple Lie algebra over an algebraically closed field of characteristic zero (e.g., ), a Cartan subalgebra is the same thing as a maximal abelian subalgebra consisting of elements x such that the
adjoint endomorphism is
semisimple (i.e.,
diagonalizable). Sometimes this characterization is simply taken as the definition of a Cartan subalgebra.[1]pg 231
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.
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]
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 (an element is semisimple if the
adjoint endomorphism induced by it is
diagonalizable). 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.[2] In that case, may be taken as the complexification of the Lie algebra of a
maximal torus of the compact group.
If is a
linear Lie algebra (a Lie subalgebra of the Lie algebra of endomorphisms of a finite-dimensional vector space V) over an algebraically closed field, then any Cartan subalgebra of is the
centralizer of a maximal
toral subalgebra of .[citation needed] 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
matrices over a field, is the algebra of all diagonal matrices.[citation needed]
For the special Lie algebra of traceless matrices , it has the Cartan subalgebra
where
For example, in the Cartan subalgebra is the subalgebra of matrices
with Lie bracket given by the matrix commutator.
The Lie algebra of by matrices of trace has two non-conjugate Cartan subalgebras.[citation needed]
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 of by matrices of trace has a Cartan subalgebra of rank but has a maximal abelian subalgebra of dimension consisting of all matrices of the form with any by 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 (or, since it is normalized by diagonal matrices).
But, it turns out these weights can be used to classify the irreducible representations of the Lie algebra . For a finite dimensional irreducible -representation , there exists a unique weight with respect to a partial ordering on . Moreover, given a such that for every positive root , there exists a unique irreducible representation . This means the root system contains all information about the representation theory of .[1]pg 240
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 (indeed, split reductive 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.
Cartan subgroup
For a Cartan subgroup of a linear algebraic group, see
Cartan subgroup.
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.
Examples of Cartan Subgroups
The subgroup in GL2(R) consisting of diagonal matrices.
Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
ISBN978-3319134666