Calabi–Yau properties of nontrivial Noetherian DG down-up algebras

Abstract

In this paper, we introduce and study differential graded (DG) down–up algebras. In brief, a DG down–up algebra [Formula: see text] is a connected cochain DG algebra such that its underlying graded algebra [Formula: see text] is a graded down–up algebra. We describe all possible differential structures on Noetherian DG down–up algebras. For those Noetherian DG down-up algebras with nonzero differential, we compute their DG automorphism groups; study their isomorphism problems; and show that they are all Calabi–Yau DG algebras.