Cartan subalgebra
From Wikipedia, the free encyclopedia
In mathematics, a Cartan subalgebra is a nilpotent subalgebra 
of a Lie algebra 
that is self-normalising (if ![[X,Y] \in \mathfrak{h}](../../../math/3/2/c/32c8a76f54da4c5a7b9c745cf8fd6878.png)
for all 
, then 
).
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 restricted to diagonalize the representation, and the eigenspace of the zero weight vector is . 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.
[edit] See also
[edit] 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
