Duality for Differential Operators of Lie-Rinehart Algebras

Thierry Lambre (1) , Patrick Le Meur (2, 1)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

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.

Domaines: Théorie des représentations [math.RT]

Thierry Lambre, Patrick Le Meur. Duality for Differential Operators of Lie-Rinehart Algebras. Pacific Journal of Mathematics, 2018, 297 (2), pp.405-454. ⟨10.2140/pjm.2018.297.405⟩. ⟨hal-02361304⟩