The projective Leavitt complex

Abstract

For a finite quiver $Q$ without sources, we consider the corresponding radical square zero algebra $A$. We construct an explicit compact generator for the homotopy category of acyclic complexes of projective $A$-modules. We call such a generator the projective Leavitt complex of $Q$. This terminology is justified by the following result: the opposite differential graded endomorphism algebra of the projective Leavitt complex of $Q$ is quasi-isomorphic to the Leavitt path algebra of $Q^{op}$. Here, $Q^{op}$ is the opposite quiver of $Q$ and the Leavitt path algebra of $Q^{op}$ is naturally $Z$-graded and viewed as a differential graded algebra with trivial differential.