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Deformations of Koszul Artin–Schelter Gorenstein algebras

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Abstract

We compute the Nakayama automorphism of a Poincaré–Birkhoff–Witt (PBW)-deformation of a Koszul Artin–Schelter (AS) 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 AS–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 article is elementary and based on linear algebra. The results obtained in this article will be applied in a subsequent paper (He et al., Skew polynomial algebras with coefficients in AS regular algebras, preprint, 2011).

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Authors and Affiliations

  1. Department of Mathematics, Shaoxing College of Arts and Sciences, Shaoxing, 312000, Zhejiang, China

    Ji-Wei He

  2. Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, 2020, Antwerp, Belgium

    Ji-Wei He & Fred Van Oystaeyen

  3. Department WNI, University of Hasselt, Universitaire Campus, 3590, Diepenbeek, Belgium

    Yinhuo Zhang

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  1. Ji-Wei He
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Correspondence to Ji-Wei He.

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He, JW., Van Oystaeyen, F. & Zhang, Y. Deformations of Koszul Artin–Schelter Gorenstein algebras. manuscripta math. 141, 463–483 (2013). https://doi.org/10.1007/s00229-012-0580-z

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  • DOI: https://doi.org/10.1007/s00229-012-0580-z

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