I talked about the Lie algebra a while back. Now I’m going to do it more properly, and using the tools developed. This is going to feature prominently in some of the proofs in the sequel.
Now, let’s see how all this works for the familiar case of , with its usual generators . This is a simple Lie algebra in fact. To see this, let’s consider the ideal of generated by some nonzero vector ; I claim it is all of .
Consider the three cases :
First, assume or is nonzero. Bracketing with , and again, gives
Using a vanderMonde invertibility of this system of linear equations, we find that either or belongs to . Say does, for definiteness; then too; from this, as well. Thus .
If , then from , we find , which implies and similarly for . Thus .
I claim now that the algebra is in fact a Cartan subalgebra. Indeed, it is easily checked to be maximal abelian. Moreover, since acts by a diagonalizable operator on the faithful representation on , it follows that is (abstractly) semisimple. (more…)