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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.

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  • $\begingroup$ 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
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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.

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