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The number $12$ (or, probably we shall say Bernoulli numbers in general) appears in many places in Mathematics, sometimes leading to unexpected connections between different topics.

For instance, some time ago there was a very interesting explanation for

1) its occurrence in the Todd class

and

2) its occurrence in the Euler-Maclaurin formula

in terms of Riemann-Roch for toric varieties, as explained in:

Euler-Maclaurin formula and Riemann-Roch

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My question is, will there be some relation between 1) and

3) its occurrence in the Baker-Campbell-Hausdorff formula.

I guess this might be related to some explicit local expressions in some method of proving the index theorem on Lie groups, or even the Duflo map (which I don't really understand).

Thank you very much.

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    $\begingroup$ Isn't this a numerology question?! At least you don't ask why 12 is close to an integer... :-) $\endgroup$
    – Wadim Zudilin
    Commented Jul 15, 2010 at 7:27
  • $\begingroup$ Hi Wadim, I edited the question make it somehow more "feasible" :-) $\endgroup$
    – Bo Peng
    Commented Jul 15, 2010 at 7:42
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    $\begingroup$ A baker's dozen is 13. I don't know about Campbell and Hausdorff. $\endgroup$
    – Tom Goodwillie
    Commented Jul 15, 2010 at 11:11
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    $\begingroup$ Possibly related mathoverflow.net/questions/9220/… $\endgroup$
    – SandeepJ
    Commented Jul 15, 2010 at 14:11
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    $\begingroup$ Have you looked at the Wikipedia article, en.wikipedia.org/wiki/Baker-Campbell-Hausdorff_formula? Please, clarify in what way the explanation there involving the expansion of $\operatorname{ad}(X)/(e^{\operatorname{ad}(X)}-1)$ found there is lacking? $\endgroup$
    – Victor Protsak
    Commented Jul 15, 2010 at 17:56

1 Answer 1

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The answer to your question is the following: given two non-commutative variables $x$ and $y$ one has $$ log(e^xe^y)=x+e^{ad_x}\frac{ad_x}{e^{ad_x}-1}(y)+O(y^2) $$

It is not the appearance of $12$ that is intriguing, but the appearance of the Todd series in algebraic geometry. It suggests that there is a group hidden somewhere... and this is indeed the case. This group is the derived loop space of your favorite algebraic variety $X$, and its tangent Lie algebra is the shifted tangent sheaf $T_X[-1]$, with Lie bracket given by the Atiyah class (the fact that the Atiyah class gives rize to a Lie structure was discovered by Kapranov).

The universal enveloping algebra of this Lie algebra is the Hochschild complex of $X$. One then gets a nice dictionnary between the Lie side and the algebraic geometry side. E.g.:

  • any object in the derived category of $X$ turns out to be a representation of this Lie algebra.

  • Poincare-Birkhoff-Witt is Hochschild-Kostant-Rosenberg.

  • the Duflo isomorphism is the Kontsevich-Caldararu isomorphism between the Harmonic and Hochschild structures.

  • there is also an relation between closed embeddings in algebraic geometry and inclusions of Lie algebras.

  • ...

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  • $\begingroup$ In item 1 "derived category of X " do you mean current sheaves on X? PS very useful answer! $\endgroup$
    – Alexander Chervov
    Commented Dec 5, 2012 at 19:14
  • $\begingroup$ Coherent sheaves $\endgroup$
    – Alexander Chervov
    Commented Dec 5, 2012 at 19:43
  • $\begingroup$ I actualy mean quasi-coherent sheaves (the action of $T_X[-1]$ on a given one $E$ is given by the Atiyah class of $E$). $\endgroup$
    – DamienC
    Commented Dec 5, 2012 at 20:07

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