Brown, K.A. and Zhang, J.J.
(2008)
Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras.
*Journal of Algebra*, 320(5),
pp. 1814-1850.
(doi:10.1016/j.jalgebra.2007.03.050)

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## Abstract

We show that many noetherian Hopf algebras A have a rigid dualising complex R with [FORMULA]. 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=S^{2}ξ, 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) Poincaré 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 S^{4} 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.

Item Type: | Articles |
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Status: | Published |

Refereed: | Yes |

Glasgow Author(s) Enlighten ID: | Brown, Professor Ken |

Authors: | Brown, K.A., and Zhang, J.J. |

Subjects: | Q Science > QA Mathematics |

College/School: | College of Science and Engineering > School of Mathematics and Statistics > Mathematics |

Journal Name: | Journal of Algebra |

ISSN: | 0021-8693 |

Published Online: | 30 June 2008 |

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