Leavitt path algebras with finitely presented irreducible representations

Let E be an arbitrary graph, K be any field and let L = L-K(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 L-K(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)]. (C) 2015 Elsevier Inc. All rights reserved.