Differential projective modules over algebras with radical square zero

TL;DR

This paper explores the structure of differential projective modules over radical square zero algebras derived from quivers, establishing a functorial relationship with quiver representations and analyzing Gorenstein properties.

Contribution

It constructs a full and dense functor linking differential projective modules over radical square zero algebras to quiver representations and investigates Gorenstein conditions for certain algebras.

Findings

Established a functor from differential projective modules to quiver representations

Proved the algebra of dual numbers over certain radical square zero algebras is not virtually Gorenstein

Analyzed conditions under which the algebra exhibits Gorenstein properties

Abstract

Let $Q$ be a finite quiver and $\Lambda$ be the radical square zero algebra of $Q$ over a field. We give a full and dense functor from the category of reduced differential projective modules over $\Lambda$ to the category of representations of the opposite of $Q$. If moreover $Q$ has oriented cycles and $Q$ is not a basic cycle, we prove that the algebra of dual numbers over $\Lambda$ is not virtually Gorenstein.