Indecomposable representations of the Kronecker quivers

TL;DR

This paper discusses the structure of indecomposable modules over the n-Kronecker algebra, providing simplified proofs for the existence of certain modules and their properties related to roots of associated Kac-Moody algebras.

Contribution

It offers a concise proof that all exceptional modules over finite quiver path algebras are tree modules, extending understanding of module classification.

Findings

Existence of tree modules for every positive root

At least n tree modules for every positive imaginary root

All exceptional modules are tree modules

Abstract

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.