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Cartan subalgebra (Definition)

Let $\fr g$ be a Lie algebra. Then a Cartan subalgebra is a maximal subalgebra of $\fr g$ which is self-normalizing, that is, if $[g,h]\in\fr h$ for all $h\in\fr h$ , then $g\in\fr h$ as well. Any Cartan subalgebra $\fr h$ is nilpotent, and if $\fr g$ is semi-simple, it is abelian. All Cartan subalgebras of a Lie algebra are conjugate by the adjoint action of any Lie group with algebra $\fr g$ .

The dimension of $\fr h$ is called the rank of $\fr g$ .




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Also defines:  rank of a Lie algebra

Attachments:
rank (Lie algebra) (Definition) by rmilson
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Cross-references: rank, dimension, algebra, Lie group, adjoint action, conjugate, abelian, semi-simple, nilpotent, self-normalizing, subalgebra, Lie algebra
There are 7 references to this entry.

This is version 4 of Cartan subalgebra, born on 2002-12-27, modified 2005-09-26.
Object id is 3849, canonical name is CartanSubalgebra2.
Accessed 5713 times total.

Classification:
AMS MSC17B20 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Simple, semisimple, reductive )

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