Representations of Leavitt Path Algebras

TL;DR

This paper explores the structure of Leavitt path algebra representations, establishing their connection to quiver representations, characterizing finite dimensional quotients, and linking algebraic properties to graph-theoretic criteria.

Contribution

It introduces a new categorical equivalence between Leavitt path algebra modules and quiver representations, and provides a graph-theoretic criterion for finite dimensional quotients.

Findings

Category of L-modules is equivalent to a subcategory of quiver representations

Necessary and sufficient graph conditions for finite dimensional quotients

L has UGN and is algebraically amenable under certain conditions

Abstract

We study representations of a Leavitt path algebra $L$ of a finitely separated digraph $\Gamma$ over a field. We show that the category of $L$-modules is equivalent to a full subcategory of quiver representations. When $\Gamma$ is a (non-separated) row-finite digraph we determine all possible finite dimensional quotients of $L$ after giving a necessary and sufficient graph theoretic criterion for the existence of a nonzero finite dimensional quotient. This criterion is also equivalent to $L$ having UGN (Unbounded Generating Number) as well as being algebraically amenable. We also realize the category of $L$-modules as a retract, hence a quotient by an explicit Serre subcategory of the category of quiver representations (that is, $\mathbb{F}\Gamma$-modules) via a new colimit model for $M\otimes_{\mathbb{F}\Gamma} L$.