# Universal enveloping algebra of the Lie algebra of vector fields on a manifold

Suppose we have a smooth manifold $M$ with structure sheaf $\mathcal{O}_M$ and $\mathfrak{X}(M)$ is the Lie algebra of vector fields on $M$. How should we interpret the universal enveloping algebra $U(\mathfrak{X}(M))$? Is it just the algebra of all (finite order) differential operators $\mathcal{O}_M \to \mathcal{O}_M$?

• Thanks! In that case, would the universal enveloping algebra $U(\mathfrak{X}(M))$ correspond to the differential operators which have constant coefficients (in any local chart)?
• No, because already the elements of $X(M)$ are not of that form! Aug 15 '16 at 17:52