On a conjecture by Pierre Cartier about a group of associators

Abstract

In \cite{cartier2}, Pierre Cartier conjectured that for any non commutative formal power series $\Phi$ on $X=\{x_0,x_1\}$ with coefficients in a $\Q$-extension, $A$, subjected to some suitable conditions, there exists an unique algebra homomorphism $\varphi$ from the $\Q$-algebra generated by the convergent polyz\^etas to $A$ such that $\Phi$ is computed from $\Phi_{KZ}$ Drinfel'd associator by applying $\varphi$ to each coefficient. We prove $\varphi$ exists and it is a free Lie exponential over $X$. Moreover, we give a complete description of the kernel of polyz\^eta and draw some consequences about a structure of the algebra of convergent polyz\^etas and about the arithmetical nature of the Euler constant.