Extensions of simple modules over Leavitt path algebras

Let E be a directed graph, K any field, and let LK(E) denote the Leavitt path algebra of E with coefficients in K. For each rational infinite path c∞ of E we explicitly construct a projective resolution of the corresponding Chen simple left LK(E)-module V[c∞]. Further, when E is row-finite, for each irrational infinite path p of E we explicitly construct a projective resolution of the corresponding Chen simple left LK(E)-module V[p]. For Chen simple modules S,T we describe ExtLK(E)1(S,T) by presenting an explicit K-basis. For any graph E containing at least one cycle, this description guarantees the existence of indecomposable left LK(E)-modules of any prescribed finite length.