Indecomposable representations of the Kronecker quivers

Let $k$ be a field and $\Lambda$ 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 $\Lambda$-modules are the positive roots of the corresponding Kac-Moody algebra. Thorsten Weist has shown that for every positive root there are 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.