
Journal of Lie Theory 25 (2015), No. 4, 11671190 Copyright Heldermann Verlag 2015 Divergence and qDivergence in Depth 2 Anton Alekseev Dept. of Mathematics, University of Geneva, 24 rue du Lièvre, 1211 Geneva 4, Switzerland Anton.Alekseev@unige.ch Anna Lachowska Dept. of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT 06511, U.S.A. Anna.Lachowska@yale.edu Elise Raphael Dept. of Mathematics, University of Geneva, 24 rue du Lièvre, 1211 Geneva 4, Switzerland Elise.Raphael@unige.ch [Abstractpdf] The KashiwaraVergne Lie algebra ${\frak krv}$ encodes symmetries of the KashiwaraVergne problem on the properties of the CampbellHausdorff series. It is conjectured that ${\frak krv} \cong {\Bbb K}t \oplus {\frak grt}_1$, where $t$ is a generator of degree 1 and ${\frak grt}_1$ is the GrothendieckTeichm\"uller Lie algebra. In this paper, we prove this conjecture in depth 2. The main tools in the proof are the divergence cocycle and the representation theory of the dihedral group $D_{12}$. Our calculation is similar to the calculation by Zagier of the graded dimensions of the double shuffle Lie algebra in depth 2.\endgraf In analogy to the divergence cocycle, we define the superdivergence and $q$divergence cocycles (here $q^l=1$) on Lie subalgebras of ${\frak grt}_1$ which consist of elements with weight divisible by l. We show that in depth 2 these cocycles have no kernel. This result is in sharp contrast with the fact that the divergence cocycle vanishes on $[{\frak grt}_1, {\frak grt}_1]$. Keywords: KashiwaraVergne conjecture, divergence cocycle, GrothendieckTeichmueller Lie algebra. MSC: 17B01, 81R10 [ Fulltextpdf (383 KB)] for subscribers only. 