Resolutions over Koszul algebras. (Q2569214)

Let \(\Lambda\) be a quotient of the path algebra of a quiver by a homogeneous (in the grading given by path length) ideal. When \(\Lambda\) is Koszul, the authors show that a minimal projective resolution of the degree zero part \(\Lambda_0\) of \(\Lambda\), viewed as a right \(\Lambda\)-module, admits a comultiplication. This has interesting consequences. First, it is shown that the above resolution allows to construct a minimal projective resolution of \(\Lambda_0\) viewed as a \textit{left} \(\Lambda\)-module. Secondly, the same resolution allows to construct a minimal projective resolution of \(\Lambda_0\) viewed as a right module over the enveloping algebra \(\Lambda^e\) of \(\Lambda\).