Cartan subalgebra

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 $\\mathfrak{g}$ .

单词	CartanSubalgebra
释义	Cartan subalgebra 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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