\"Uber erreichbare und baumartige unzerlegbare Darstellungen von K\"ochern

TL;DR

This paper investigates the construction of indecomposable modules over finite-dimensional algebras using tree-representations, proving key results about exceptional representations and their subrepresentations, with applications to the 3-Kronecker quiver.

Contribution

It provides a detailed proof that exceptional representations are tree-representations and introduces a functor for constructing indecomposables from bipartite quiver representations.

Findings

Exceptional representations are tree-representations.

Each exceptional representation of a generalized Kronecker quiver has an indecomposable sub- or factor-representation.

Small dimension calculations demonstrate the accessibility of 3-Kronecker-quiver representations.

Abstract

Let $A$ be a finite-dimensional algebra over an algebraically closed field. The problem of constructing indecomposable $A$-modules inductively from simple ones by means of exact sequences - called accessibility - is the starting point of the present diploma-thesis. It has lead us to the consideration of exceptional and indecomposable tree-representations of finite quivers. Following Ringel, we prove his result that exceptional representations are tree-representations. We give a detailed description of the various aspects of the Schofield-Induction which plays an important role in the proof. Moreover we introduce a functor (strong hypotheses being given) which enables us to construct indecomposable modules of an algebra from indecomposable representations of a certain bipartite quiver. We also give a proof of Ringel's result that each exceptional representation of dimension $d>1$ of a generalized Kronecker quiver has an indecomposable factor- or subrepresentation of dimension $d-1$. The thesis is concluded by some calculations showing the accessibility of representations of the 3-Kronecker-quiver in small dimensions.