OK, now we’ve gotten some of the basic facts about the root space decomposition down. So, as usual is a semisimple Lie algebra and a Cartan subalgebra; we have the decomposition , where is the root system. For each , we can choose a pair of vectors . Then generate a subalgebra which is isomorphic to . Here and is a multiple of , which in turn is the dual to under the Killing form that identifies .

That was a lightning review of where we are; if you’ve missed something, check back at this post.

The notation suggests that the algebra should only depend on and not on the particular choice of (but is uniquely determined from and ). Indeed, this is the case, and it follows from

Proposition 1When , is one-dimensional.

Choose any coming from suitable and . We have a representation of on

(recall ) and we can apply the representation theory of to it. (more…)