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Let $G$ be a compact semisimple Lie group. I have found to different definitions of its rank: One of them defined the rank of the Lie group to be the dimension of a maximal torus. The other definition defined the rank to be the dimension of a Cartan subalgebra in the Lie algebra. Do these definitions coincide?

Niklas
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1 Answers1

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For semisimple Lie algebras, a Cartan subalgebra is a maximal abelian subalgebra, i.e., a maximal torus. So the definitions coincide.

Dietrich Burde
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  • It seems that "semisimple" is irrelevant here. See Théorème 2 (a), Section 2.2, Chapitre 9 of Bourbaki's Groupes et Algèbres de Lie. – Doug Mar 15 '22 at 10:06
  • @DougLiu In general, a Cartan subalgebra is not a maximal abelian subalgebra, but a maximal nilpotent subalgebra (consider a nilpotent Lie algebra). For semisimple, these two notions coincide. This is what I meant. – Dietrich Burde Mar 15 '22 at 10:19
  • You are definitely right. But do we need "semisimple" for the OP's question? – Doug Mar 15 '22 at 11:19
  • I see. No. But I was assuming he only considers the semisimple case. – Dietrich Burde Mar 15 '22 at 11:43