Appendix C: Cartan Subalgebras

Abstract

Suppose that L is a semisimple Lie algebra with Cartan subalgebra H and associated root system Ф. We want to show that if H ₁ is another Cartan subalgebra of L, with associated root system Ф ₁, then Ф ₁ is isomorphic to Ф. This shows first of all that the root system of a semisimple Lie algebra is well-defined (up to isomorphism) and secondly that semisimple Lie algebras with different root systems cannot be isomorphic.

The general proof of this statement is quite long and difficult and requires several ideas which we have so far avoided introducing. So instead we give a proof that assumes that L is a classical Lie algebra. This will be sufficient to show that the only isomorphisms between the classical Lie algebras come from isomorphisms between their root systems; we used this fact at the end of Chapter 12. We then show how Serre’s Theorem and the classification of Chapter 14 can be used to give the result for a general semisimple Lie algebra.

We conclude by discussing the connection between our Cartan subalgebras and the “maximal toral algebras” used by other authors, such as Humphreys [14].