Deformations of Koszul Artin-Schelter Gorenstein algebras

TL;DR

This paper investigates how certain algebraic deformations affect the Calabi-Yau and Gorenstein properties of Koszul algebras, providing explicit automorphisms and criteria for preserving these properties.

Contribution

It introduces explicit formulas for Nakayama automorphisms of PBW-deformations and establishes criteria for when deformations retain Calabi-Yau properties.

Findings

Explicit Nakayama automorphisms for dimensions 2 and 3

Criteria for PBW-deformations to be Calabi-Yau

Deformations defined by a potential under mild conditions

Abstract

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.