  We compute the Nakayama automorphism of a PBW-deformation of a Koszul
Artin-Schelter Gorenstein algebra of finite global dimension, and give a
criterion for an augmented PBW-deformation of a Koszul Calabi-Yau algebra to be
Calabi-Yau. The relations between the Calabi-Yau property of augmented
PBW-deformations and that of non-augmented cases are discussed. The Nakayama
automorphisms of PBW-deformations of Koszul Artin-Schelter Gorenstein algebras
of global dimensions 2 and 3 are given explicitly. We show that if a
PBW-deformation of a graded Calabi-Yau algebra is still Calabi-Yau, then it is
defined by a potential under some mild conditions. Some classical results are
also recovered. Our main method used in this paper is elementary and based on
linear algebra. The results obtained in this paper will be applied in a
subsequent paper.
