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# Cartan Subalgebra

Let be a finite-dimensional Lie algebra over some field . A subalgebra of is called a Cartan subalgebra if it is nilpotent and equal to its normalizer, which is the set of those elements such that .

It follows from the definition that if is nilpotent, then itself is a Cartan subalgebra of . On the other hand, let be the Lie algebra of all endomorphisms of (for some natural number ), with . Then the set of all endomorphisms of of the form is a Cartan subalgebra of .

It can be proved that:

1. If is infinite, then has Cartan subalgebras.

2. If the characteristic of is equal to , then all Cartan subalgebras of have the same dimension.

3. If is algebraically closed and its characteristic is equal to 0, then, given two Cartan subalgebras and of , there is an automorphism of such that .

4. If is semisimple and is an infinite field whose characteristic is equal to 0, then all Cartan subalgebras of are Abelian.

Every Cartan subalgebra of a Lie algebra is a maximal nilpotent subalgebra of . However, a maximal nilpotent subalgebra of doesn't have to be a Cartan subalgebra. For instance, if is the Lie algebra of all endomorphisms of with and if is the subalgebra of all endomorphisms of the form , then is a maximal nilpotent subalgebra of , but not a Cartan subalgebra.

Cartan Algebra, Subalgebra

This entry contributed by José Carlos Santos

## References

Bourbaki, N. Ch. 7-9 in Lie Groups and Lie Algebras. New York: Springer-Verlag, 2005.Jacobson, N. Lie Algebras. New York: Dover, 1979.

## Referenced on Wolfram|Alpha

Cartan Subalgebra

## Cite this as:

Santos, José Carlos. "Cartan Subalgebra." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CartanSubalgebra.html