# Leavitt path algebras with at most countably many irreducible representations

### Pere Ara

Universitat Autònoma de Barcelona, Bellaterra, Spain### Kulumani M. Rangaswamy

University of Colorado at Colorado Springs, USA

## Abstract

Let $E$ be an arbitrary directed graph with no restrictions on the number of vertices and edges and let $K$ be any field. We give necessary and sufficient conditions for the Leavitt path algebra $L_{K}(E)$ to be of countable irreducible representation type, that is, we determine when $L_{K}(E)$ has at most countably many distinct isomorphism classes of simple left $L_{K}(E)$-modules. It is also shown that $L_{K}(E)$ has finitely many isomorphism classes of simple left modules if and only if $L_{K}(E)$ is a semi-artinian von Neumann regular ring with finitely many ideals. Equivalent conditions on the graph $E$ are also given. Examples are constructed showing that for each (finite or infinite) cardinal $κ$ there exists a Leavitt path algebra $L_{K}(E)$ having exactly $κ$ distinct isomorphism classes of simple right modules.

## Cite this article

Pere Ara, Kulumani M. Rangaswamy, Leavitt path algebras with at most countably many irreducible representations. Rev. Mat. Iberoam. 31 (2015), no. 4, pp. 1263–1276

DOI 10.4171/RMI/868