Duality for Differential Operators of Lie-Rinehart Algebras
arXiv:1709.03973 · doi:10.2140/pjm.2018.297.405
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.
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