Irreducible representations of Leavitt algebras

Abstract

For a weighted graph E, we construct representation graphs F, and consequently, LK(E)-modules VF, where LK(E) is the Leavitt path algebra associated to E, with coefficients in a field K. We characterise representation graphs F such that VF are simple LK(E)-modules. We show that the category of representation graphs of E, RG(E), is a disjoint union of subcategories, each of which contains a unique universal object T which gives an indecomposable LK(E)-module VT and a unique irreducible representation graph S, which gives a simple LK(E)-module VS.Specialising to graphs with one vertex and m loops of weight n, we construct irreducible representations for the celebrated Leavitt algebras LK(n,m). On the other hand, specialising to graphs E with weight one, we recover the simple modules of Leavitt path algebras LK(E) constructed by Chen via infinite paths or sinks and give a large class of non-simple indecomposable LK(E)-modules.Our approach gives a completely new way to construct indecomposable and simple modules for Leavitt path algebras of (weighted) graphs. Besides being more visual, this approach allows for carrying calculus on these modules with ease. On the other hand, the approach also allows us, to the best of our knowledge, for the first time, to produce systematically many examples of non-simple indecomposable modules, for these algebras, including Leavitt algebras LK(n,m).