Stably Calabi-Yau properties of derivation quotient algebras

The aim of this paper is to study bimodule stably Calabi-Yau properties of derivation quotient algebras. We give the definition of a twisted stably Calabi-Yau algebra and show that every twisted derivation quotient algebra $A$ for which the associated bimodule complex gives the beginning of a bimodule resolution for $A$ is bimodule stably twisted Calabi-Yau. In this setting we give a new interpretation of some results by Yu [Yu12], implying that $A$ is almost Koszul of periodic type. Using the characterization of higher preprojective algebras given by Amiot and Oppermann in [AO14], we prove that finite dimensional bigraded derivation quotient algebras with homogeneous potential and exact associated complex are higher preprojective algebras of their degree-zero subalgebra, which is Koszul and $(d-1)$-representation finite.