Cartan subalgebra

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In mathematics, a Cartan subalgebra is a nilpotent subalgebra $\mathfrak{h}$ of a Lie algebra $\mathfrak{g}$ that is self-normalising (if $[X,Y] \in \mathfrak{h}$ for all $X \in \mathfrak{h}$ , then $Y \in \mathfrak{h}$ ).

Cartan subalgebras exist for finite dimensional Lie algebras whenever the base field is infinite. If the field is algebraically closed of characteristic 0 and the algebra is finite dimensional then all Cartan subalgebras are conjugate under automorphisms of the Lie algebra, and in particular are all isomorphic.

A Cartan subalgebra of a finite dimensional semisimple Lie algebra over an algebraically closed field of characteristic 0 is abelian and also has the following property of its adjoint representation: the weight eigenspaces of $\mathfrak{g}$ restricted to $\mathfrak{h}$ diagonalize the representation, and the eigenspace of the zero weight vector is $\mathfrak{h}$ . The non-zero weights are called the roots, and the corresponding eigenspaces are called root spaces, and are all 1-dimensional.

Kac-Moody algebras and generalized Kac-Moody algebras also have Cartan subalgebras.

The name is for Élie Cartan.

[edit] Examples

Any nilpotent Lie algebra is its own Cartan subalgebra.
A Cartan subalgebra of the Lie algebra of n×n matrices over a field is the algebra of all diagonal matrices.
The Lie algebra sl₂(R) of 2 by 2 matrices of trace 0 has two non-conjugate Cartan subalgebras.
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 sl_2n(C) of 2n by 2n matrices of trace 0 has rank 2n−1 but has a maximal abelian subalgebra of dimension n² consisting of all matrices of the form ${0, A\choose 0, 0}$ with A any n by n 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 (which is also not a Cartan subalgebra since it is normalized by diagonal matrices).