Lie-Rinehart and Poisson algebras over $C^{\infty }$-rings

Abstract

We define the analogue of Lie-Rinehart algebras over $C^{\infty }$-rings. We show that given a Poisson $C^{\infty }$-ring $\mathcal{A}$ its module ${\Omega }_{\mathcal{A}}^1$ of $C^{\infty }$-Kähler differentials is (part of) a Lie-Rinehart algebra. Conversely, given a Lie-Rinehart algebra $\mathcal{M}\stackrel{\rho }{\to }{C}^{\infty }\mathrm{Der}(\mathcal{A})$ over a $C^{\infty }$-ring $\mathcal{A}$, there is a natural Poisson bracket on the $C^{\infty }$-ring $\mathcal{F}(\mathcal{M})$ associated with the $\mathcal{A}$-module $\mathcal{M}$ (the $C^{\infty }$-ring analogue of an $\mathcal{A}$-algebra freely generated by the module $\mathcal{M}$). In the case where $\mathcal{A}$ is the $C^{\infty }$-ring of smooth functions on a manifold $M$ and $\mathcal{M}$ is the module $\Gamma (E)$ of sections of a Lie algebroid $E\to M$, the $C^{\infty }$-ring $\mathcal{F}(\Gamma (E))$ is the ring of functions $C^{\infty }(E^{\vee })$ on the total space of the vector bundle $E^{\vee }\to M$ dual to the vector bundle $E$.

62 pages. v3: added a reference and updated another reference. v2: addition of Examples 2.3 and 2.11. Comments welcome