The Kac-Moody algebras are divided in three very distinct classes: finite-dimensional, affine and indefinite type.
- For the first class the finite-dimensional representation theory is very known.
- For the second one, it is not hard to see that we only have trivial representation.
Question: does exists finite-dimensional representations of any Kac-Moody algebra of indefinite type which is not trivial?
EDIT: After I posted the question, I realized that I know the answer, so I posted it.
The answer is NO. There is no finite-dimensional representation theory for indefinite type, since they are all simple.