Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras

By Kenneth A. Brown, James J. Zhang. 2006

168 citations recorded by the source corpus.

We show that many noetherian Hopf algebras A have a rigid dualising complex R with R≅A1ν[d]. Here, d is the injective dimension of the algebra and ν is a certain k-algebra automorphism of A, unique up to an inner automorphism. In honour of the finite-dimensional theory which is hereby generalised we call ν the Nakayama automorphism of A. We prove that ν=S2ξ, where S is the antipode of A and ξ is the left winding automorphism of A determined by the left integral of A. The Hochschild homology and cohomology groups with coefficients in a suitably twisted free bimodule are shown to be non-zero in the top dimension d, when A is an Artin–Schelter regular noetherian Hopf algebra of global dimension d. (Twisted) Poincare duality holds in this setting, as is deduced from a theorem of Van den Bergh. Calculating ν for A using also the opposite coalgebra structure, we determine a formula for S4 generalising a 1976 formula of Radford for A finite-dimensional. Applications of the results to the cases where A is PI, an enveloping algebra, a quantum group, a quantised function algebra and a group algebra are outlined.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3270b4c01334c0224c73094fb371b04aa6a48ff0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. James J. Zhang
2. U. Krähmer
3. Kenneth A. Brown
4. D. Rogalski
5. Manuel Reyes
6. Q.-S. Wu
7. S.-Q. Wang
8. L.-Y. Liu

In Geometry and Topology

1. James J. Zhang
2. U. Krähmer
3. Kenneth A. Brown
4. D. Rogalski
5. Manuel Reyes
6. Q.-S. Wu
7. S.-Q. Wang
8. L.-Y. Liu

In Statistical and Nonlinear Physics

1. U. Krähmer
2. Manuel Reyes
3. S.-Q. Wang
4. L.-Y. Liu
5. D. Rogalski
6. Q.-S. Wu
7. James J. Zhang
8. Kenneth A. Brown

Citations in the corpus

• Prime regular Hopf algebras of GK‐dimension one (2008, background): “Recent years have seen substantial progress in our understanding of (infinite dimensional) noetherian Hopf algebras [3, 5, 12, 23].”
• Derived Categories (2016, unknown): “Zhang [32] introduced rigid Gorenstein algebras.”
• Twisted Calabi-Yau property of Ore extensions (2012, methodology): “eralized to a more general situation — the PBW deformations of Koszul Calabi-Yau algebras [WZ11]. The quantized enveloping algebra of a complex semisimple Lie algebra is always Calabi-Yau [Che04]. In [BZ08], Brown and Zhang also described the Nakayama automorphism explicitly by using homological int”
• Calabi-Yau Frobenius algebras (2007, background): “However, it might be interesting to try to apply this formalism to finite-dimensional Hopf algebras (analogously to [BZ06]).”
• Skew Calabi–Yau algebras and homological identities (2013, methodology): “es (E2.8.1). Conversely, if (E2.8.1) holds, then applying the winding automorphism Ξr γ◦S to both sides of the previous displayed equation and then applying 1#ǫgives (E2.8.2). Remark 2.9. Recall from [BZ] that the Nakayama automorphism of a noetherian AS Gorenstein Hopf algebraHhas the form φ= S−2◦Ξ”
• On the Hochschild (co)homology of quantum homogeneous spaces (2008, methodology): “If B = A and ε is Gorenstein, then Brown and Zhang’s result also says that ω ≃ Aσ for some σ ∈ Aut(A) [5], by which we mean it is isomorphic to A as left module but the right action is given by a ◭ b := aσ(b).”