The projective Leavitt complex

TL;DR

This paper constructs an explicit compact generator called the projective Leavitt complex for the homotopy category of acyclic complexes of projective modules over a radical square zero algebra derived from a finite quiver, linking it to Leavitt path algebras.

Contribution

It introduces the projective Leavitt complex as a new explicit generator and establishes a quasi-isomorphism with the Leavitt path algebra of the opposite quiver.

Findings

The projective Leavitt complex is a compact generator for the homotopy category.

The endomorphism algebra of this complex is quasi-isomorphic to the Leavitt path algebra.

This links the homotopy category of complexes to Leavitt path algebras via explicit constructions.

Abstract

Let Q be a finite quiver without sources, and A be the corresponding algebra with radical square zero. 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.