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 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.

Contents

Examples

See also

References

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.

See 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