The Kashiwara-Vergne Problem as an Excuse




The Kashiwara-Vergne conjecture is a subtle property of the Campbell-Hausdorff series which implies that the Duflo isomorphism theorem can be extended to the germs of bi-invariant distributions on a Lie group. This problem is related to the combinatorics of multiple zeta values and to the Deligne-Drinfeld conjecture. It is also related to some open questions in deformation quantization. An important corollary of the Kashiwara-Vergne conjecture was proved in papers of M. Andler, A. Dvorsky, S. Sahi, and C. Torossian. For an arbitrary Lie algebra the conjecture was established by A. Alekseev and E. Meinrenken. A. Alekseev and C. Torossian built a nice algebraic framework for the Kashiwara-Vergne problem. Using this framework they showed that solutions of the Kashiwara-Vergne problem are closely related to Drinfeld's associators. In fact, starting from any Drinfeld associator one can produce a solution of the Kashiwara-Vergne problem. In our seminar we will talk about the Alekseev-Torossian algebraic framework for the generalized Kashiwara-Vergne problem, we will talk about the Grothendieck-Teichmueller Lie algebra and Drinfeld's associators. Finally we will discuss Kontsevich's graphical calculus and the Alekseev-Torossian connection.

The seminar is based on the sequence of papers

If you are a graduate student and you are taking this seminar for credit then you need to give at least one talk or solve a set of homework problems. I am going to post the homework problems in the middle of the quarter.

Please, e-mail me if you want to give a talk.