Finite Dimensional Representations of Leavitt Path Algebras

Abstract

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

21 pages. The acknowledgements of our grant support did not appear in the previous version. arXiv admin note: substantial text overlap with arXiv:1510.03382