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 $\mathcal{O}_M \to \mathcal{O}_M$?


It isn't, because it does not contain the operators given by multiplication by a non-constant function.

You can view the Lie algebra as (part of) a Lie-Rinehart algebra, and then the corresponding enveloping algebra is indeed that of finite order differential operators.

  • 1
    $\begingroup$ 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)? $\endgroup$
    – ಠ_ಠ
    Aug 15 '16 at 9:37
  • 2
    $\begingroup$ No, because already the elements of $X(M)$ are not of that form! $\endgroup$ Aug 15 '16 at 17:52

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