Finite Dimensional Representations of Quivers with Oriented Cycles

TL;DR

This paper explores the representation theory of truncated path algebras of quivers with cycles, analyzing how their finite-dimensional modules behave as the truncation level increases.

Contribution

It provides both old and new insights into the representation theory of these truncated algebras and their connection to modules over the original path algebra.

Findings

Characterization of finite-dimensional modules over truncated algebras

Development of methods to analyze module categories as truncation level varies

Enhanced understanding of the relationship between truncated and original path algebra modules

Abstract

Let $K$ be a field, $Q$ a quiver, and $\mathcal{A}$ the ideal of the path algebra $KQ$ that is generated by the arrows of $Q$. We present old and new results about the representation theories of the truncations $KQ/\mathcal{A}^L$, $L \in \mathbb{N}$, tracking their development as $L$ goes to infinity. The goal is to gain a better understanding of the category of those finite dimensional $KQ$-modules which arise as finitely generated modules over admissible quotients of $KQ$.