# Revista Matemática Iberoamericana

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**Volume 31, Issue 4, 2015, pp. 1263–1276**

**DOI: 10.4171/RMI/868**

Published online: 2015-12-23

Leavitt path algebras with at most countably many irreducible representations

Pere Ara^{[1]}and Kulumani M. Rangaswamy

^{[2]}(1) Universitat Autònoma de Barcelona, Bellaterra, Spain

(2) University of Colorado at Colorado Springs, USA

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 $\kappa$ there exists a Leavitt path algebra $L_{K}(E)$ having exactly $\kappa$ distinct isomorphism classes of simple right modules.

*Keywords: *Leavitt path algebra, irreducible representation, socle, von Neumann regular

Ara Pere, Rangaswamy Kulumani: Leavitt path algebras with at most countably many irreducible representations. *Rev. Mat. Iberoamericana* 31 (2015), 1263-1276. doi: 10.4171/RMI/868