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Let $\mathcal{M}$ be a smooth real manifold and let $A:= \mathcal{C}\left(\mathcal{M}\right)$ be the real algebra of smooth functions on $\mathcal{M}$.

Recall from McConnell, Robson, Noncommutative Noetherian rings; chapter 15 the following definitions

  • The derivation ring $\Delta(A)$ of $A$ is the subalgebra of $\mathrm{End}_{\mathbb{R}}(A)$ generated by $A$ (considered as left translations) and $\mathrm{Der}_{\mathbb{R}}(A)$.
  • The ring of differential operators $\mathcal{D}(A)$ of $A$ (also referred to as the Grothendieck's algebra of differential operators: Is there a “categorical” description of Grothendieck's algebra of differential operators?) is defined recursively as follows $$\mathcal{D}_{-1}(A) = 0 \qquad \text{and} \\ \mathcal{D}_k(A) = \left\{\partial \in \mathrm{End}_{\mathbb{R}}(A)\mid [\partial,a] \in \mathcal{D}_{k-1}(A)\right\} \qquad \text{for all} \quad k\geq 0, \\ \text{where} \qquad [\partial,a](b) = \partial(ab) - a\partial(b).$$

Recall also the following two additional facts/definitions

  • The triple $(A,\mathrm{Der}_\mathbb{R}(A),\mathrm{id}_{\mathrm{Der}_\mathbb{R}(A)})$ is a Lie-Rinehart algebra and hence we can consider its universal enveloping algebra $U_A\left(\mathrm{Der}_\mathbb{R}(A)\right)$ (see, for instance, page 7 of Huebschmann, Poisson cohomology and quantization)
  • A differential operator of order $\leq k$ on the smooth manifold $\mathcal{M}$ is an $\mathbb{R}$-linear map $D:\mathcal{C}(\mathcal{M}) \to \mathcal{C}(\mathcal{M})$ such that in local coordinates it looks like $D f (p)= \sum_{i_1+...+i_n \leq k} A_{i_1 ... i_n}(p) \frac{\partial^{i_1+...+i_n} f}{\partial x_1^{i_1} ...x_n^{i_n}}|_p$ for some smooth functions $A_{i_1 ... i_n}$ (see, e.g., Differential operators as sections of a vector bundle and references therein). Denote by $D(A)$ the algebra of differential operators on $\mathcal{M}$ of any order.

In Nestruev, Smooth manifolds and observables; Theorem 9.62 it is proved that $D(A) = \mathcal{D}(A)$. By construction/universal property one has canonical maps $$U_A\left(\mathrm{Der}_{\mathbb{R}}(A)\right) \twoheadrightarrow \Delta(A) \hookrightarrow \mathcal{D}(A) = D(A).$$ It seems to me that it is folklore in differential geometry that these four objects are, in fact, the same (that is to say, that the two canonical maps are isomorphisms). See, for instance, Huebschmann, Poisson cohomology and quantization; page 8 or Xu, Quantum Groupoids; Example 3.1.

Question: Can anybody suggest to me a (good) reference where this has been proven in details?

Related (unanswered) questions: Map of the enveloping algebra to the ring of differential operators, The algebra generated by derivations.

Comments: I already asked this question on MSE, but probably that was not the correct place for such a technical topic.

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