Simple Modules over Leavitt Path Algebras Generated by Indicator Functions on Paths

Abstract

Let E be an arbitrary graph, K a field, and D the set of boundary paths of E consisting of both infinite paths and finite paths ending at singular vertices. The space KD of K-valued functions on D is a right module over the Leavitt path algebra LK(E). For a non- empty set S ? D that is strongly connected and closed under adding and deleting initial segments, we study the submodule N_S = 1_S.LK(E) generated by the indicator function of S. Let S_0 be the set of vertices lying on the paths in S. We show that N_S is simple if and only if S contains no breaking vertex of the hereditary saturated set H = E0 nS0. In particular, when S contains only infinite paths or when E is a row- finite graph, NS is a simple module. Up to isomorphism, the simple module NS depends only on S_0. When S is an infinite set of paths, the simple module NS is neither graded nor projective. The left module LK(E)1_S, obtained by twisting the action through the involution of LK(E), is simple exactly when NS is, and, when S is infinite, it is not isomorphic to any of the Chen simple modules. For the rose graph R_n with n ? 2 petals, we show that LK(R_n)1_S is isomorphic to one of the twisted Chen modules constructed by Nam, Srivastava and Vien, and to none of the other simple modules in their list. Moreover, when S is not a single vertex, the simple module NS is finitely presented if and only if only finitely many edges of E have both source and range in S_0; in particular, every simple module N_S is finitely presented when E is a finite graph. Finally, we determine when the analogous module generated by an indicator function over the Cohn path algebra CK(E) is simple. 1.