The injective Leavitt complex

TL;DR

This paper constructs an explicit compact generator called the injective Leavitt complex for the homotopy category of acyclic complexes of injective modules over a finite dimensional algebra derived from a quiver without sinks, linking it to Leavitt path algebras.

Contribution

It introduces the injective Leavitt complex as a new explicit generator and establishes its endomorphism algebra's quasi-isomorphism to the Leavitt path algebra.

Findings

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

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

Provides an explicit construction connecting quiver algebras to Leavitt path algebras.

Abstract

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