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Cartan subalgebra (Definition)

Let $ \mathfrak{g}$ be a Lie algebra. Then a Cartan subalgebra is a maximal subalgebra of $ \mathfrak{g}$ which is self-normalizing, that is, if $ [g,h]\in\mathfrak{h}$ for all $ h\in\mathfrak{h}$, then $ g\in\mathfrak{h}$ as well. Any Cartan subalgebra $ \mathfrak{h}$ is nilpotent, and if $ \mathfrak{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 $ \mathfrak{g}$.

The dimension of $ \mathfrak{h}$ is called the rank of $ \mathfrak{g}$.



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Also defines:  rank of a Lie algebra

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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 4566 times total.

Classification:
AMS MSC: 17B20 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Simple, semisimple, reductive )
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