The Kac Theorem

Let A be hereditary.
A yields a symmetrizable Cartan datum, thus a Kac-Moody Lie-algebra g of rank n = n(A). (In case A is representation-finite, g is just a fin.dim. semisimple complex Lie-algebra).

We recall: a Kac-Moody Lie-algebra has a triangular decomposition

and n+ can further be decomposed:

 
here Φ+ is the set of positive roots, and  
It is the set Φ+ of positive roots which is now of interest:
We return to mod A and consider its Grothendieck group K0(A) = Zn, again n = n(A).
For an A-module M, let dim M be the corresponding element in K0(A)
(dim M is called the "dimension vector" of M,
it counts the multiplicities of the various simple modules as composition factors of M).

Kac Theorem. The map dim furnishes a surjection
from the class of indecomposable A-modules onto Φ+.