Finite Dimensional Representations of Leavitt Path Algebras

TL;DR

This paper classifies all finite dimensional modules of Leavitt path algebras associated with row-finite directed graphs using a combinatorial reduction algorithm, revealing their tame representation type.

Contribution

It provides an explicit Morita equivalence and a combinatorial algorithm to classify finite dimensional modules of Leavitt path algebras for row-finite graphs.

Findings

Finite dimensional modules are classified via a reduction algorithm.

The category of finite dimensional modules is tame.

Finite dimensional modules differ in complexity from quiver representations.

Abstract

When $\Gamma$ is a row-finite di(rected )graph we classify all finite dimensional modules of the Leavitt path algebra $L(\Gamma)$ via an explicit Morita equivalence given by an effective combinatorial (reduction) algorithm on the digraph $\Gamma$. The category of (unital) $L(\Gamma)$-modules is equivalent to a subcategory of quiver representations of $\Gamma$. However the category of finite dimensional representations of $L(\Gamma)$ is tame in contrast to the finite dimensional quiver representations of $\Gamma$ which are almost always wild.