Severa's GRT Question
January-31-11
11:46 PM
In Drinfeld's Gal(\bar Q/Q) paper there is a bijection between associators and elements of the Lie algebra gt with the numerical component equal to 1 (gt(K) is composed of pairs (s,psi), where s\in K and psi is from the completed free Lie algebra with 2 generators, satisfying some (5gon&6gon) equations; in the bijection we demand s=1). The bijection psi<->Phi ((1,psi) in gt, Phi an associator) is given by the equation (saying that (1,psi) acts on Phi by rescaling) 
                          Phi(x,y)^-1 d/dt Phi(tx,ty)|_{t=1} = psi(x, Phi^-1 y Phi).   (*)
The question is: do we know some psi "explicitly"? For example, what are the coefficients of psi_KZ?
Maybe a better formulation of the question: Let's just suppose that Phi is a group-like element of K<<x,y>>, but let's forget about 5gon and 6gon. Then (*) defines a Lie series psi. Since Phi is group-like, we can express the product of any of its coefficients as a linear combination of some other coefficients. As a result, the coefficients of psi are linear combinations of coefficients of Phi. Is there some intelligent way how to write these linear combinations?
(the reason for asking is that the 5gon & 6gon relations for psi are (inhomogeneous) linear, unlike the 5gon & 6gon for Psi; seeing that they are just the double shuffle equations would be nice. Anyway, psi is "the right log of Phi" (log Phi being "the wrong log of Phi"))