# Cartan subalgebra

Let $\mathfrak{g}$ be a Lie algebra. Then a 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 $\mathfrak{g}$.

Title Cartan subalgebra CartanSubalgebra 2013-03-22 13:20:09 2013-03-22 13:20:09 bwebste (988) bwebste (988) 7 bwebste (988) Definition msc 17B20 rank of a Lie algebra