Leavitt path algebras with finitely presented irreducible representations

Let E be an arbitrary graph, K   be any field and let L=LK(E)L=LK(E) be the corresponding Leavitt path algebra. Necessary and sufficient conditions (both graphical and algebraic) are given under which all the irreducible representations of L are finitely presented. In this case, the graph E turns out to be row-finite and the cycles in E form an artinian partial ordered set under a defined relation ≥. When the graph is E is finite, the above graphical conditions were shown in [6] to be equivalent to LK(E)LK(E) having finite Gelfand–Kirillov dimension. Examples are constructed showing that this equivalence no longer holds for infinite graphs and a complete description is obtained of Leavitt path algebras over arbitrary graphs having finite Gelfand–Kirillov dimensions. The “building blocks” for these algebras seem to be von Neumann rings and the Laurent polynomial ring K[x,x−1]K[x,x−1].