Guillaume Laplante-Anfossi: Kashiwara—Vergne solutions degree by degree

Abstract:

The original Kashiwara–Vergne problem was posed in the context of convolutions on Lie groups in 1978, and has wide implications from Lie theory to harmonic analysis. It was reformulated in 2012 by Alekseev–Torossian, who show that a KV solution is an automorphism of the degree completed free Lie algebra on two generators, satisfying two equations. In this talk, I will explain how Kashiwara—Vergne solutions can be extended degree by degree. This can be used to simplify the computation of a class of Drinfel’d associators, which under the Alekseev–Torossian conjecture, may comprise all associators. I will also give a proof that the associated graded Lie algebra of the Kashiwara–Vergne group is isomorphic to the graded Kashiwara-Vergne Lie algebra, a result which should be compared to an analogous, but more difficult to prove, statement about the Grothendieck—Teichmüller group. This is joint work with Zsuzsanna Dancso, Iva Halacheva and Marcy Robertson.