Let $X$ be an algebraic variety (or a complex manifold) over $\mathbb C$. Let $D(X)$ be its algebra of differential operators. Mostly I am interested in algebraic differential operators, but the case of smooth coefficients is also interesting(though the computation of Hochschild/cyclic (co)homology for these options coincide).

What is known about cohomology of $D(X)$ of this algebra viewed as a Lie algebra?

I know how to compute cyclic and Hohschild (co)homology of $D(X)$ and there is a theorem of Loday and Quillen which relates Lie algebra homology of $\mathfrak{gl}_{\infty}(A)$ to cyclic homology of $A$ for any algebra $A$, but Lie algebra homology of $A$ and $\mathfrak{gl}_{\infty}(A)$ may differ greately as some cycles may die on the infinite level.

  • $\begingroup$ Could you say what is $A$? And how do you define differential operator: with polynomial/analytic/smooth/continuous coefficients? $\endgroup$ – YCor Mar 17 '16 at 8:59
  • $\begingroup$ (+ fix the typo "Lle algebra", "HoChschild" and missing space before "(or complex manifold)" , then I'll erase this comment) $\endgroup$ – YCor Mar 17 '16 at 9:01
  • $\begingroup$ ... and no idea what you mean by "homology of $A$" $\endgroup$ – YCor Mar 17 '16 at 12:51
  • $\begingroup$ @YCor Thank you for you comments, tried to make the question clearer $\endgroup$ – lks8271 Mar 17 '16 at 20:14
  • $\begingroup$ you left "Hohschild", it's "Hochschild", and you now say "the Lie algebra homology of $A$" which I don't understand ($A$ is meant to be an associative algebra) $\endgroup$ – YCor Mar 17 '16 at 22:02

Lie algebra (co)homology of the Lie algebra of differential operators was used in the Feigin-Tsygan works on the Riemann-Roch theorem. See, for example, this paper and later papers of Tsygan with co-authors.

| cite | improve this answer | |

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.