On a conjecture by Pierre Cartier about a group of associators

TL;DR

This paper proves Cartier's conjecture on associators, showing the existence and uniqueness of a homomorphism relating polyzeta values to a given formal power series, and explores implications for polyzeta algebra structure and Euler constant.

Contribution

It establishes the existence and uniqueness of the algebra homomorphism conjectured by Cartier, providing a detailed description of the kernel and algebraic structure involved.

Findings

Proves Cartier's conjecture on associators.

Describes the kernel of polyzeta algebra.

Analyzes the arithmetical properties of the Euler constant.

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.