On a conjecture by Pierre Cartier

Details On a conjecture by Pierre Cartier On a conjecture by Pierre Cartier

2009-10-10

arxiv.org/abs/0910.1932v9

Mathematics

Combinatorics

Scientific paper

In \cite{cartier2}, Pierre Cartier conjectured that for any 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 $\phi$ 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 $\phi$ to each coefficient. We prove that $\phi$ exists and that it is a free Lie exponential over $X$. Moreover, we give the complete description of the kernel of the polyz\^etas and draw some consequences about a structure of the algebra of polyz\^etas and about the arithmetical nature of the Euler constant.

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