  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.
