Nilpotent subalgebra of a Lie algebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra formula_1 of a Lie algebra formula_2 that is self-normalising (if formula_3 for all formula_4, then formula_5). They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra formula_2 over a field of characteristic formula_7.
In a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero (e.g., formula_8), a Cartan subalgebra is the same thing as a maximal abelian subalgebra consisting of elements "x" such that the adjoint endomorphism formula_9 is semisimple (i.e., diagonalizable). Sometimes this characterization is simply taken as the definition of a Cartan subalgebra.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.
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).
Existence and uniqueness.
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 formula_10 over an algebraically closed field of characteristic zero, there is a simpler approach: by definition, a toral subalgebra is a subalgebra of formula_10 that consists of semisimple elements (an element is semisimple if the adjoint endomorphism induced by it is diagonalizable). A Cartan subalgebra of formula_10 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, formula_1 may be taken as the complexification of the Lie algebra of a maximal torus of the compact group.
If formula_2 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 formula_2 is the centralizer of a maximal toral subalgebra of formula_2. If formula_2 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 formula_10 is semisimple, then the adjoint representation presents formula_10 as a linear Lie algebra, so that a subalgebra of formula_10 is Cartan if and only if it is a maximal toral subalgebra.
Cartan subalgebras of semisimple Lie algebras.
For finite-dimensional semisimple Lie algebra formula_10 over an algebraically closed field of characteristic 0, a Cartan subalgebra formula_44 has the following properties:
These two properties say that the operators in formula_47 are simultaneously diagonalizable and that there is a direct sum decomposition of formula_2 as
formula_50
where 
formula_51.
Let formula_52. Then formula_53 is a root system and, moreover, formula_54; i.e., the centralizer of formula_1 coincides with formula_1. The above decomposition can then be written as:
formula_57
As it turns out, for each formula_58, formula_59 has dimension one and so:
formula_60.
See also Semisimple Lie algebra#Structure for further information.
Decomposing representations with dual Cartan subalgebra.
Given a Lie algebra formula_2 over a field of characteristic formula_32, and a Lie algebra representationformula_63 there is a decomposition related to the decomposition of the Lie algebra from its Cartan subalgebra. If we set
formula_64
with formula_65, called the weight space for weight formula_66, there is a decomposition of the representation in terms of these weight spaces formula_67 In addition, whenever formula_68 we call formula_69 a weight of the formula_2-representation formula_71.
Classification of irreducible representations using weights.
But, it turns out these weights can be used to classify the irreducible representations of the Lie algebra formula_2. For a finite dimensional irreducible formula_2-representation formula_71, there exists a unique weight formula_58 with respect to a partial ordering on formula_76. Moreover, given a formula_58 such that formula_78 for every positive root formula_79, there exists a unique irreducible representation formula_80. This means the root system formula_53 contains all information about the representation theory of 
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 formula_1 then it is called "splittable," and the pair formula_83 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.
A Cartan subgroup of a Lie group is a special type of subgroup. Specifically, its Lie algebra (which captures the group’s algebraic structure) is itself a Cartan subalgebra. 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. There are multiple definitions for a Cartan subgroup. 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
References.
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