On a conjecture by Pierre Cartier about a group of associators

Abstract

In [8], Pierre Cartier conjectured that for any non commutative formal power series Φ on X = {x0, x1} with coefficients in a Q-extension, A, subjected to some suitable conditions, there exists an unique algebra homomorphism ϕ from the Q-algebra generated by the convergent polyzêtas to A such that Φ is computed from ΦKZ Drinfel'd associator by applying ϕ to each coefficient. We prove ϕ exists and it is a free Lie exponential over X. Moreover, we give a complete description of the kernel of polyzêta and draw some consequences about a structure of the algebra of convergent polyzêtas and about the arithmetical nature of the Euler constant.

Dual bases for noncommutative symmetric and quasi-symmetric functions via monoidal factorization

“…By (52), we get, for any n ≥ 1, log Y | t n = π 1 (y n ) and since {π 1 (y n )} n≥1 generates freely Prim(H ) [13], the expected result follows.…”

“…The noncommutative symmetric function S 1 = Λ 1 is primitive but {S n } n≥2 and {Λ n } n≥2 are neither primitive nor group-like. Moreover, by (13), (14) and (15), one has…”

Mathematical Renormalization in Quantum Electrodynamics via Noncommutative Generating Series

“…the monoidal factorization facilitates mainly the renormalization and the computation of the associators 4 via the universal one, i.e. Φ KZ of Drinfel'd [44].…”

“…Moreover, Z 1 , Z 2 are equal and stand for the noncommutative generating series of {ζ (w)} w∈Y * −y 1 Y * , or {ζ (w)} w∈x 0 X * x 1 , as one has Z 1 = Z 2 = π Y Z ⊔⊔ [43,44,45]. This allows, by extracting the coefficients of the noncommutative generating series, to explicit the counter-terms eliminating the divergence of {Li w } w∈x 1 X * and of {H w } w∈y 1 Y * and this leads naturally to an equation connecting algebraic structures…”

“…In this paper, we will focus on the approach by noncommutative formal power series, adapted from [22,23], and explain how the results of [43,44,45], allow to study the combinatorial aspects of the renormalization at the singularities in {0, 1, +∞} of the solutions of linear differential equations (see Example 1 below) as well as the solutions of nonlinear differential equations (see Examples 2 and 3 below) described in Section 3.2 and involved in quantum electrodynamics.…”