Poincaré–Birkhoff–Witt deformations of Calabi–Yau algebras

Abstract

Recently, Bocklandt proved a conjecture by Van den Bergh in its graded version, stating that a graded quiver algebra A (with relations) which is Calabi–Yau of dimension 3 is defined from a homogeneous potential W. In this paper, we prove that if we add to W any potential of smaller degree, we get a Poincaré–Birkhoff–Witt deformation of A. Such PBW deformations are Calabi–Yau and are characterised among all the PBW deformations of A. Various examples are presented.