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In Cartier (Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents. Sém. BOURBAKI, 53ème 2000–2001, no. 885), Pierre Cartier conjectured that for any non-commutative formal power series Φ on X={x 0,x 1} with coefficients in a \(\mathbb{Q}\)-extension, A, subjected to some suitable conditions, there exists a unique algebra homomorphism φ from the \(\mathbb{Q}\)-algebra generated by the convergent polyzetas to A such that Φ is computed from the Φ KZ Drinfel’d associator by applying φ to each coefficient. We prove that φ exists and that it is a free Lie exponential map over X. Moreover, we give a complete description of the kernel of ζ and draw some consequences about the arithmetical nature of the Euler constant and about an algebraic structure of the polyzetas.

On a conjecture by Pierre Cartier about a group of associators