Smash products of Calabi–Yau algebras by Hopf algebras

Abstract

Let H be a Hopf algebra and A be an H-module algebra. This article investigates when the smash product A♯H is (skew) Calabi-Yau, has Van den Bergh duality or is Artin-Schelter regular or Gorenstein. In particular, if A and H are skew Calabi-Yau, then so is A♯H and its Nakayama automorphism is expressed using the ones of A and H. This is based on a description of the inverse dualising complex of A♯H when A is a homologically smooth dg algebra and H is homologically smooth and with invertible antipode. This description is also used to explain the compatibility of standard constructions of Calabi-Yau dg algebras with taking smash products.Note that it is proved in [15, Theorem 2.3] that, when H is noetherian, H is Calabi-Yau if and only if S 2 is an inner automorphism and H is Artin-Schelter regular with trivial left homological integral.This article is based on the description of RHom Λ e (Λ, Λ e ). When S is invertible, there exists a dg A-bimodule D A which is H S 2 -equivariant in the sense of [30] (see Section 4) and such that D A ≃ RHom A e (A, A e ) in D(A e ). A suitable extension of D A is then isomorphic to RHom Λ e (Λ, Λ e ) in the following sense. See 5.5.1 for a general statement. See also [12,14,15,16,26,30,36,37] for previous results describing RHom Λ e (Λ, Λ e ) when A is a connected graded algebra and H is finitedimensional, semisimple or cocommutative.Proposition 2 (5.5.2). Let H be a Hopf algebra with Van den Bergh duality in dimension d. Let A be an H-module dg algebra. Assume that A is homologically smooth. Then, Λ is homologically smooth and RHom Λ e (Λ,The description of RHom Λ e (Λ, Λ e ) can be used to describe the deformed Calabi-Yau completions of Λ. Recall that HH n−2 (A) ≃ H 0 Hom A e (D A [n − 1], A[1]) when D A is cofibrant over A e , which is possible to assume. The following result was proved in [25] when H is the (semisimple) group algebra of a finite group.

A note on the bijectivity of the antipode of a Hopf algebra and its applications

“…Recently, Meur showed that, by imposing a purely homological restriction, any twisted Calabi-Yau Hopf algebra has bijective antipode [11,Proposition 1]. The next result proved in the present paper uses both homological and ring-theoretic restrictions on a Hopf algebra.…”

Cohomology of Graded Twisting of Hopf Algebras

“…Then, the isomorphism Ext i A e (A, A e ) ∼ = Ext i A (k A , A A ) ⊗ A is actually an isomorphism of the right kΓ-modules. Then, one can check that the above definition coincides with the (weak) homological determinant defined in [20].…”

“…For a more detailed account on the actions of Hopf algebras on algebras, we refer to the book by Montgomery [18] and the paper by Centrone [19]. Although the description of the Hochschild cohomology of A Γ can be derived from the results in [20], we give a complete and more direct proof for the results needed. Previous results about the cohomology of crossed products can also be found, for example, in [21][22][23][24] and the references therein.…”

“…The homological determinant of a Hopf action on a connected AS-Gorenstein algebra is already defined in [24,31,32]. In [20], the author defined the (weak) homological determinant of a Hopf action on a twisted CY algebra. Let A be a Hopf algebra as in the above definition.…”

Skew Calabi-Yau property of normal extensions

“…We concern with the general situation, not-necessarily connected, for skew Calabi-Yau property. Interesting examples of such algebras arise naturally as factor algebras of quiver algebras [2,3], and as smash products of (connected graded) algebras by Hopf algebras [11].…”