Cartan subalgebra

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 andalso 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.

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"_2"n"(C) of 2"n" by 2"n" matrices of trace 0 has rank 2"n"−1 but has a maximal abelian subalgebra of dimension "n"² consisting of all matrices of the form $\{0,\; Achoose\; 0,\; 0\}$ 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).

ee also

*Cartan subgroup
*Carter subgroup

References

*Nathan Jacobson, "Lie algebras", ISBN 0-486-63832-4
*J.E. Humphreys, "Introduction to Lie Algebras and Representation Theory", ISBN 0-387-90053-5
*springer|id=C/c020550|first=V.L.|last= Popov