Representation theory of hereditary artin algebras of finite representation type

TL;DR

This paper classifies the structure of modules over hereditary artin algebras of finite type, providing methods to construct their Auslander-Reiten quivers and compute related invariants.

Contribution

It determines all hammocks in the Auslander-Reiten quiver of such algebras, enabling effective construction and analysis of module categories.

Findings

Complete classification of hammocks in the Auslander-Reiten quiver.

Method for constructing the Auslander-Reiten quiver from the ext-quiver.

Computed numbers of indecomposable modules and nilpotency degrees.

Abstract

Let $H$ be a hereditary artin algebra of finite representation type. We first determine all hammocks in the Auslander-Reiten quiver $\GaH$ of $\mmod H$, the category of finitely generated left $H$-modules. This enables us to obtain an effective method to construct $\GaH$ by simply viewing the ext-quiver of $H$. As easy applications, we compute the numbers of non-isomorphic indecomposable objects in $\mmod H$ and the associated cluster category $\mathscr{C}_H$, as well as the nilpotencies of the radicals of $\mmod H\hspace{-.4pt},$ $\hspace{-.5pt} D^{\hspace{.5pt}b\hspace{-.6pt}}(\hspace{-.5pt}\mmod H\hspace{-.5pt})$ and $\mathscr{C}_H$.