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.