# The projective Leavitt complex

Huanhuan Li
Western Sydney University, Sydney, Australia
31 October, 12 noon, Carslaw 373, University of Sydney

## 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.