Duality for Differential Operators of Lie-Rinehart Algebras
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
Cite
@article{arxiv.1709.03973,
title = {Duality for Differential Operators of Lie-Rinehart Algebras},
author = {Thierry Lambre and Patrick Le Meur},
journal= {arXiv preprint arXiv:1709.03973},
year = {2018}
}
Comments
Correction of typos and of an argument in the proof of Lemma 5.1.2. Minor changes to stick to the published version. Published in the Pacific Journal of Mathematics