They were introduced by Élie Cartan in his doctoral thesis.
It controls the representation theory of a semi-simple Lie algebra
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.
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
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.
may be taken as the complexification of the Lie algebra of a maximal torus of the compact group.
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
over an algebraically closed field of characteristic 0, a Cartan subalgebra
has the following properties: (As noted earlier, a Cartan subalgebra can in fact be characterized as a subalgebra that is maximal among those having the above two properties.)
are simultaneously diagonalizable and that there is a direct sum decomposition of
has dimension one and so: See also Semisimple Lie algebra#Structure for further information.
,[clarification needed] and a Lie algebra representation
, there is a decomposition of the representation in terms of these weight spaces
But, it turns out these weights can be used to classify the irreducible representations of the Lie algebra
[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
Over a non-algebraically closed field not every semisimple Lie algebra is splittable, however.
When we consider the identity component of a subgroup, it shares the same Lie algebra.
However, there isn’t a universally agreed-upon definition for which subgroup with this property should be called the ‘Cartan subgroup,’ especially when dealing with disconnected groups.
For compact connected Lie groups, a Cartan subgroup is essentially a maximal connected Abelian subgroup—often referred to as a ‘maximal torus.’ The Lie algebra associated with this subgroup is also a Cartan subalgebra.
Now, when we explore disconnected compact Lie groups, things get interesting.
One common approach, proposed by David Vogan, defines it as the group of elements that normalize a fixed maximal torus while preserving the fundamental Weyl chamber.
This version is sometimes called the ‘large Cartan subgroup.’ Additionally, there exists a ‘small Cartan subgroup,’ defined as the centralizer of a maximal torus.
It’s important to note that these Cartan subgroups may not always be abelian in genera