On a conjecture by Pierre Cartier about a group of associators

In \cite{cartier2}, Pierre Cartier conjectured that for any non commutative formal power series $\mathrm{\Phi }$ on $X=\{x_0,x_1\}$ with coefficients in a $\text{Q}$-extension, $A$, subjected to some suitable conditions, there exists an unique algebra homomorphism $\phi$ from the $\text{Q}$-algebra generated by the convergent polyz^etas to $A$ such that $\mathrm{\Phi }$ is computed from ${\mathrm{\Phi }}_{KZ}$ Drinfel’d associator by applying $\phi$ to each coefficient. We prove $\phi$ 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.