Skew Calabi–Yau algebras and homological identities

Abstract

Abstract. A skew Calabi-Yau algebra is a generalization of a Calabi-Yau algebra which allows for a non-trivial Nakayama automorphism. We prove three homological identities about the Nakayama automorphism and give several applications. The identities we prove show (i) how the Nakayama automorphism of a smash product algebra A#H is related to the Nakayama automorphisms of a graded skew Calabi-Yau algebra A and a finite-dimensional Hopf algebra H that acts on it; (ii) how the Nakayama automorphism of a graded twist of A is related to the Nakayama automorphism of A; and (iii) that the Nakayama automorphism of a skew Calabi-Yau algebra A has trivial homological determinant in case A is noetherian, connected graded, and Koszul. IntroductionWhile the Calabi-Yau property originated in geometry, it now has incarnations in the realm of algebra that seem to be of growing importance. Calabi-Yau triangulated categories were introduced by Kontsevich [Ko] in 1998. See [Ke] for an introductory survey about Calabi-Yau triangulated categories. Calabi-Yau algebras were introduced by Ginzburg [Gi] in 2006 as a noncommutative version of coordinate rings of Calabi-Yau varieties. Since the late 1990s, the study of CalabiYau categories and algebras has been related to a large number of other topics such as quivers with superpotentials, DG algebras, cluster algebras and categories, string theory and conformal field theory, noncommutative crepant resolutions, and Donaldson-Thomas invariants. Some fundamental questions in the area were answered by Van den Bergh recently in [VdB3].One known method for constructing a noncommutative Calabi-Yau algebra is to form the smash product A#H of a Calabi-Yau algebra A with a Calabi-Yau Hopf algebra H that acts nicely on A. This phenomenon has been studied quite broadly; for instance, see [BSW], [Fa, Section 3] We will employ the following notation. Let A be an algebra over a fixed commutative base field k. The unmarked tensor ⊗ always means ⊗ k . Let M be an A-bimodule, and let µ, ν be algebra automorphisms of A. Then µ M ν denotes the induced A-bimodule such that µ M ν = M as a k-space, and where Definition 0.1. Let A be an algebra over k.(a) A is called skew Calabi-Yau (or skew CY, for short) if (i) A is homologically smooth, that is, A has a projective resolution in the category A e -Mod that has finite length and such that each term in the projective resolution is finitely generated, and (ii) there is an integer d and an algebra automorphism µ of A such that Calabi-Yau and µ A is inner (or equivalently, µ A can be chosen to be the identity map after changing the generator of the bimodule 1 A µ ).There is some variation in the literature concerning the exact definition of (skew) CY algebras. Ginzburg included in his definition of a CY algebra [Gi, Definition 3.2.3] the condition that the A e -projective resolution of A is self-dual, but Van den Bergh has shown that this is satisfied automatically [VdB3, Appendix C]. We are also not the first to study skew CY algebras; the notion has been calle...

PBW deformations of Artin–Schelter regular algebras

“…This idea was formalized recently in [25]. On the other hand, (skew) Calabi-Yau algebras need not be graded.…”

Automorphisms for Some Symmetric Multiparameter Quantized Weyl Algebras and Their Localizations

“…As a result, A q,Γ n (K) is strongly cancellative and cancellative. ✷ Recall from [25,29] that a K−algebra A is called µ−twisted (or skew) Calabi-Yau of dimension d, where µ is a K−algebra automorphism of A and d ∈ Z ≥0 , provided that…”

Spectral sequences for the cohomology rings of a smash product

“…Smash products have also appeared as a means of untwisting, or unbraiding, certain twisted structures. For example, one can untwists a twisted Calabi-Yau algebra with a smash product [12,6], or unbraid a braided Hopf algebra [3,Section 1.5]. In a more classical context, smash products have played an integral role in a classification program proposed by Andruskiewitsch and Schneider, which began at [2].…”

“…If we let Z = φ act on k q [x, y] by the automorphism φ : x → q −1 x, y → qy, then, according to [12,Proposition 7.3] and [6], the smash product k q [x, y]#Z will be Calabi-Yau. By way of Theorem 6.5, we can provide the following computation.…”