Understanding the Role of Abelian Cartan Subalgebras in Simple Lie Algebras

[topsquark]

Say we have a simple Lie Algebra, and let's use A1: \(\displaystyle [H, E^{\pm} ] = \pm 2 E^{\pm}\) and \(\displaystyle [E^+, E^- ]\) as an example. My text seems to be telling me that we can write this in a decomposition: \(\displaystyle \{ H \} \oplus g\) as H is an (Abelian) Cartan subalgebra of A1.

My question is this: \(\displaystyle E^{\pm}\) can individually be taken as Abelian Cartan subalgebras as well. That would make A1 isomorphic to \(\displaystyle \{ H \} \oplus \{ E^+ \} \oplus \{ E^- \}\), which clearly isn't correct. How do we know to single out H for special treatment?

-Dan

 

[topsquark]

Okay. I finally finished my little project and I just can't see where the text is getting this. A1 (or \(\displaystyle SL(2, \mathbb{C} )\), whichever you please, is clearly a simple Lie Algebra, not a semi-simple one. The text is working with a Cartan Weyl basis, Cartan matrix, root system, etc, all properties of a semi-simple Algebra as if A1 were semi-simple. After a long slog of picky details I finished working out a general basis for A1 and have finally proved that A1 is not isomorphic to the direct sum of a Cartan subalgebra and its other generators, and thus is not semi-simple.

So am I not understanding something (and going half-mad) or is the text wrong to do this?

-Dan