Finite-dimensional representations of Leavitt path algebras

Abstract

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