Anatomy of an associator

Abstract

We study some Lie algebras defined by solutions to the double shuffle equations with poles and construct families of explicit solutions to these equations in all weights and depths. These provide universal coordinates in which to write down `zeta elements': the images of generators of the Lie algebra of the motivic Galois group of mixed Tate motives over the integers. We expect that a similar statement holds for associators. In particular, these coordinates encode algebraic relations between multiple zeta values, and enable one to compress the currently used tables for relations between multiple zeta values in, for example, weights $\leq 13$, already by a factor of a thousand. The Lie algebras and groups studied here form part of a large algebraic structure which is related to the work of Ecalle on the calculus of moulds, and also related to the theory of universal mixed elliptic motives, and modular forms for the full modular group.