Finite Dimensional Representations of Leavitt Path Algebras

When $\mathrm{\Gamma }$ is a row-finite di(rected )graph we classify all finite dimensional modules of the Leavitt path algebra $L(\mathrm{\Gamma })$ via an explicit Morita equivalence given by an effective combinatorial (reduction) algorithm on the digraph $\mathrm{\Gamma }$. The category of (unital) $L(\mathrm{\Gamma })$-modules is equivalent to a subcategory of quiver representations of $\mathrm{\Gamma }$. However the category of finite dimensional representations of $L(\mathrm{\Gamma })$ is tame in contrast to the finite dimensional quiver representations of $\mathrm{\Gamma }$ which are almost always wild.