1
$\begingroup$

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

$\endgroup$
5
$\begingroup$

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.

$\endgroup$
2
  • 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.