Vol. 297, No. 2, 2018

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Duality for differential operators of Lie–Rinehart algebras

Thierry Lambre and Patrick Le Meur

Vol. 297 (2018), No. 2, 405–454
Abstract

Let (S,L) be a Lie–Rinehart algebra over a commutative ring R. This article proves that, if S is flat as an R-module and has Van den Bergh duality in dimension n, and if L is finitely generated and projective with constant rank d as an S-module, then the enveloping algebra of (S,L) has Van den Bergh duality in dimension n + d. When, moreover, S is Calabi–Yau and the d-th exterior power of L is free over S, the article proves that the enveloping algebra is skew Calabi–Yau, and it describes a Nakayama automorphism of it. These considerations are specialised to Poisson enveloping algebras. They are also illustrated on Poisson structures over two- and three-dimensional polynomial algebras and on Nambu–Poisson structures on certain two-dimensional hypersurfaces.

Keywords
Lie–Rinehart algebra, enveloping algebra, Calabi–Yau algebra, skew Calabi–Yau algebra, Van den Bergh duality
Mathematical Subject Classification 2010
Primary: 16E35, 16E40, 16S32, 16W25
Secondary: 17B63, 17B66
Milestones
Received: 9 January 2018
Revised: 6 June 2018
Accepted: 17 July 2018
Published: 20 December 2018
Authors
Thierry Lambre
Laboratoire de Mathématiques Blaise Pascal
UMR6620 CNRS
Université Clermont Auvergne
Campus des Cézeaux
3 place Vasarely
63178 Aubière cedex
France
Patrick Le Meur
Laboratoire de Mathématiques Blaise Pascal
UMR 6620 CNRS
Université Clermont Auvergne
Campus des Cézeaux
3 place Vasarely
63178 Aubière cedex
France
Université Paris Diderot
Sorbonne Université, CNRS
Institut de Mathématiques de Jussieu-Paris Rive Gauche
IMJ-PRG, F-75013
Paris
France