Duality for Differential Operators of Lie-Rinehart Algebras

TL;DR

This paper establishes duality properties for enveloping algebras of Lie-Rinehart algebras, extending Van den Bergh duality and exploring Calabi-Yau conditions, with applications to Poisson and Nambu-Poisson structures.

Contribution

It proves Van den Bergh duality for enveloping algebras of Lie-Rinehart algebras under specific conditions and characterizes when these algebras are skew-Calabi-Yau, including explicit automorphisms.

Findings

Enveloping algebra has Van den Bergh duality in dimension n+d.

Under Calabi-Yau conditions, the algebra is skew-Calabi-Yau.

Applications to Poisson and Nambu-Poisson structures on polynomial 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.