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⁺.

For any real positive root d, there is just one isomorphism class [M] with dim M = d.
For any imaginary positive root d, there are infinitely many isomorphism class [M] with dim M = d (provided k is infinite).