Indecomposable representations of the Kronecker quivers

Abstract

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.