Cartan subalgebra

In mathematics, especially in the theory of Lie algebras, Cartan subalgebras are used inter alia in the classification of semisimple Lie algebras and in the theory of symmetric spaces. The rank of a Lie algebra (or the associated Lie group ) is defined as the dimension of the Cartan subalgebra. An example of a Cartan subalgebra is the algebra of diagonal matrices.

Definition

It is a Lie algebra. A subalgebra is a Cartan subalgebra if it is nilpotent and selbstnormalisierend, ie when

• A and

Applies.

Examples

A Cartan subalgebra of

Is the algebra of diagonal matrices

Every Cartan subalgebra can be conjugated.

In contrast, two non- conjugate Cartan subalgebras, namely

And

Existence and uniqueness

A finite-dimensional Lie algebra over an infinite field always has a Cartan subalgebra.

For a finite-dimensional Lie algebra over an algebraically closed field Cartan subalgebras are all conjugate to each other.

Properties

If a semisimple Lie algebra over an algebraically closed field, then any Cartan subalgebra is abelian and the restriction of the adjoint representation is simultaneously diagonalizable on with the eigenspace to the weight. That is, there exists a decomposition

With

And

In the example

(when the elementary matrix with entry at the site and entries designated otherwise )

With for