Cartan subalgebra

Let 𝔤 be a Lie algebraMathworldPlanetmath. Then a Cartan subalgebraMathworldPlanetmath is a maximal subalgebraMathworldPlanetmath of 𝔤 which is self-normalizing, that is, if [g,h]𝔥 for all h𝔥, then g𝔥 as well. Any Cartan subalgebra 𝔥 is nilpotentPlanetmathPlanetmathPlanetmathPlanetmath, and if 𝔤 is semi-simplePlanetmathPlanetmath, it is abelianMathworldPlanetmath. All Cartan subalgebras of a Lie algebra are conjugate by the adjoint action of any Lie group with algebra 𝔤.

The dimension of 𝔥 is called the rank of 𝔤.

Title Cartan subalgebra
Canonical name CartanSubalgebra
Date of creation 2013-03-22 13:20:09
Last modified on 2013-03-22 13:20:09
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 7
Author bwebste (988)
Entry type Definition
Classification msc 17B20
Defines rank of a Lie algebra