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 $Ω_{\mathcal{A}}^{1}$ of $C^\infty$-Kähler differentials is (part of) a Lie-Rinehart algebra. Conversely, given a Lie-Rinehart algebra $\mathcal{M} \xrightarrowρ C^\infty\mathrm{Der}(\mathcal{A})$ 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 $Γ(E)$ of sections of a Lie algebroid $E \to M$, the $C^\infty$-ring $\mathcal{F}(Γ(E))$ 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 $E$.