The purpose of this post is to show that the category of finite-dimensional representations of a semismple Lie algebra is a semisimple category; there is thus an analogy with Maschke’s theorem, except in this case the proofs are more complicated. They can be simplified somewhat if one uses the cohomology of Lie algebras (i.e., appropriate Ext groups), which I may talk more about, but most likely only later. Here we will give the proofs based on linear algebra.
The first step is to construct certain central elements in the enveloping algebra.
Let be a nondegenerate invariant bilinear form on the Lie algebra . (E.g. could be semisimple and the Killing form.) Given a basis , we can consider the dual basis with respect to it, i.e. such that . Consider the Casimir element
I claim that is independent of the choice of and is in the center of the enveloping algebra. First off, consider the isomorphisms of -modules,
The last one is given by the form .
Now the identity, an invariant element of , is sent to . Since there is a well-defined homomorphism of vector spaces,
we see that is unique. Moreover, we can make into a -module by the adjoint—or, equivalently, commutator—mapping, i.e. for . Then becomes a -homomorphism, because
So is then an invariant element under this action of on , which means that is in the center of .
Lemma 1 Let be a semisimple subalgebra. Let be the bilinear form via . Then is nondegenerate on .
Now is the form associated to the representation so is invariant. The kernel of the form is thus an ideal , and by the second version of Cartan’s solvability criterion, is solvable. This proves the lemma. One half the proof of Cartan’s semisimplicity criterion can be generalized, as it shows.
Fix a semisimple Lie algebra over a field of characteristic zero. Let be a simple -representation, i.e. containing no proper subrepresentations besides zero.
Lemma 2 (Raising of invariants) Consider an exact sequence of -modules
where is acted upon trivially. Then it splits.
We prove this by induction on . If is not simple, then we can take a smaller submodule and consider the sequence
We get by the inductive hypothesis a -section , whose image is a one-dimensional submodule . Then .
There is another exact sequence
from which we can take a section , whose image satisfies . In particular, .
So we may assume simple, and work with our exact sequence as before.
We may make one further reduction, namely to assume that the action of on is faithful. If is the kernel of this action, it is an ideal—and, as a direct summand in , semisimple itself (as a Lie algebra). I claim that this ideal acts trivially on . Now, sends into because it acts trivially on . Since , it follows that must act trivially on . So we can treat our sequence as a sequence of -modules, where this quotient is a semisimple Lie algebra (as was a direct factor).
We have now made the reduction of the previous lemma to the following claim:
Lemma 3 (Special case) Let be a faithful, simple representation of the semisimple Lie algebra and consider an exact sequence
We have the nondegenerate bilinear form and the corresponding Casimir element . Then acts trivially on ; so . Moreover, I claim . Indeed, is a -endomorphism of , and once we prove it is nonzero, it will follow that it is an isomorphism. But
since we are in characteristic zero.
As a result, we take a nonzero vector annihilated by as the splitting. Then , so is invariant.
Finally, we can do the general case:
Theorem 4 (Weyl) Let be a -submodule of the finite-dimensional representation of the semisimple Lie algebra . Then has a -complement.
(I.e. the category is semisimple.)
We can assume that is simple (i.e. contains no proper subrepresentations). It then follows by induction that every finite-dimensional -module is a semisimple module, which implies the theorem.
So, there is an exact sequence of -modules
We consider the submodule of consisting of multiples of the identity and the inverse image ; this is then a -submodule as well. There is an exact sequence
By the previous result on raising invariants, we can find a 1-dimensional -submodule of which maps isomorphically onto . Note that any 1-dimensional -module is the trivial (action by zero) one because . This 1-dimensional module is generated by some -invariant which restricts to a nonzero multiple of the identity on . An appropriate multiple gives a -invariant projection and thus a complement.
This result is hugely important, as we will see in the future.