Lie algebras

Cartan subalgebra

In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra of a Lie algebra that is self-normalising (if for all , then ). They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra over a field of characteristic . 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.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). (Wikipedia).

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Related pages

Lie group | Algebraically closed field | Representation theory of semisimple Lie algebras | Linear Lie algebra | Diagonalizable matrix | Subalgebra | Maximal torus | Splitting Cartan subalgebra | Automorphism | Élie Cartan | Isomorphism | Split Lie algebra | Regular element of a Lie algebra | Root system | Lie algebra representation | Identity component | Abelian Lie algebra | Kac–Moody algebra | Harish-Chandra isomorphism | Mathematics | Field (mathematics) | Weyl group | Square matrix | Toral subalgebra | Lie algebra | Nilpotent Lie algebra | Semisimple operator | Semisimple Lie algebra | Generalized Kac–Moody algebra