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

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.