Eulerian idempotent and Kashiwara-Vergne conjecture

Abstract

By using the interplay between the Eulerian idempotent and the Dynkin idempotent, we construct explicitly a particular symmetric solution (F, G) of the first equation of the Kashiwara-Vergne conjecture x + y - log(e yex) = (1 - e-ad x)F(x,y) + (ead y - 1)G(x,y). Then, we explicit all the solutions of the equation in the completion of the free Lie algebra generated by two indeterminates x and y thanks to the kernel of the Dynkin idempotent.