  Let k be a field and A the n-Kronecker algebra, this is the path algebra of
the quiver with 2 vertices, a source and a sink, and n arrows from the source
to the sink. It is well-known that the dimension vectors of the indecomposable
A-modules are the positive roots of the corresponding Kac-Moody algebra.
Thorsten Weist has shown that for every positive root there are even tree
modules with this dimension vector and that for every positive imaginary root
there are at least n tree modules. Here, we present a short proof of this
result. The considerations used also provide a calculation-free proof that all
exceptional modules over the path algebra of a finite quiver are tree modules.

Representation Theory