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

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)

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=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) 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 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.

• Item Type: Articles
• 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