N-Koszul algebras, Calabi-Yau algebras and skew PBW extensions (2017)
Abstract. In the schematic approach to non-commutative algebraic geometry arises some important classes of non-commutative algebras like Koszul algebras, Artin-Schelter regular algebras, Calabi-Yau algebras, and closely related with them, the skew PBW extensions. There exist some relations between these algebras and the skew PBW extensions. We give conditions to guarantee that skew PBW extensions over fields are nonhomogeneous Koszul or Koszul algebras. We also show that a constant skew PBW extension of a field is a PBW deformation of its homogeneous version. We define graded skew PBW extensions, study some properties of these algebras and showed that if R is a PBW algebra then a graded skew PBW extension of R is a PBW algebra, and therefore, a Koszul algebra. As a generalization of the above results, we prove that every graded skew PBW extension of a finitely presented Koszul algebra is Koszul. Artin-Schelter regularity and the skew Calabi-Yau condition are studied for graded skew PBW extensions. We prove that every graded quasi-commutative skew PBW extension of an Artin-Schelter regular algebra is an Artin-Schelter regular algebra and, more general, graded skew PBW extensions of a finitely presented Auslander-regular algebra, are Artin-Schelter regular algebras. As a consequence, every graded quasi-commutative skew PBW extension of a finitely presented skew Calabi-Yau algebra is skew Calabi-Yau, and graded skew PBW extensions of a finitely presented Auslander-regular algebra are skew Calabi-Yau. Since graded quasi-commutative skew PBW extensions with coefficients in a finitely presented skew Calabi-Yau algebra are skew Calabi-Yau, the Nakayama automorphism exists for these extensions. With this in mind, we give a description of Nakayama automorphism for these non-commutative algebras using the Nakayama automorphism of the ring of the coefficients.