Todd class and Baker-Campbell-Hausdorff, or the curious number $12$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T13:02:48Z http://mathoverflow.net/feeds/question/31972 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/31972/todd-class-and-baker-campbell-hausdorff-or-the-curious-number-12 Todd class and Baker-Campbell-Hausdorff, or the curious number $12$ Bo Peng 2010-07-15T07:23:15Z 2012-12-05T10:05:26Z <p>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.</p> <p>For instance, some time ago there was a very interesting explanation for</p> <p>1) its occurrence in the Todd class</p> <p>and</p> <p>2) its occurrence in the Euler-Maclaurin formula</p> <p>in terms of Riemann-Roch for toric varieties, as explained in:</p> <p><a href="http://mathoverflow.net/questions/10667/euler-maclaurin-formula-and-riemann-roch" rel="nofollow">http://mathoverflow.net/questions/10667/euler-maclaurin-formula-and-riemann-roch</a></p> <p>===========================================================</p> <p>My question is, will there be some relation between 1) and </p> <p>3) its occurrence in the Baker-Campbell-Hausdorff formula.</p> <p>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).</p> <p>Thank you very much.</p> http://mathoverflow.net/questions/31972/todd-class-and-baker-campbell-hausdorff-or-the-curious-number-12/115488#115488 Answer by DamienC for Todd class and Baker-Campbell-Hausdorff, or the curious number $12$ DamienC 2012-12-05T10:05:26Z 2012-12-05T10:05:26Z <p>The answer to your question is the following: given two non-commutative variables $x$ and $y$ one has $$log(e^xe^y)=x+e^{ad_x}\frac{ad_x}{e^{ad_x}-1}(y)+O(y^2)$$</p> <p>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 <a href="http://arxiv.org/abs/alg-geom/9704009" rel="nofollow">discovered by Kapranov</a>). </p> <p>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.: </p> <ul> <li><p>any object in the derived category of $X$ turns out to be a representation of this Lie algebra. </p></li> <li><p>Poincare-Birkhoff-Witt is Hochschild-Kostant-Rosenberg. </p></li> <li><p>the Duflo isomorphism is the Kontsevich-Caldararu isomorphism between the Harmonic and Hochschild structures. </p></li> <li><p>there is also an relation between closed embeddings in algebraic geometry and inclusions of Lie algebras. </p></li> <li><p>...</p></li> </ul>