Leavitt path algebras with at most countably many irreducible representations

© 2015 European Mathematical Society. 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 LK(E) to be of countable irreducible representation type, that is, we determine when LK(E) has at most countably many distinct isomorphism classes of simple left LK(E)-modules. It is also shown that LK(E) has finitely many isomorphism classes of simple left modules if and only if LK (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 LK(E) having exactly κ distinct isomorphism classes of simple right modules.