#### 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 $\left(S,L\right)$ 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 $\left(S,L\right)$ 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