Leavitt path algebras with at most countably many irreducible representations

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