Mould theory and the double shuffle Lie algebra structure

TL;DR

This paper demonstrates how Racinet's theorem on the double shuffle Lie algebra structure of formal multiple zeta values can be derived simply from Ecalle's deep combinatorial and algebraic theory, clarifying complex prior results.

Contribution

It provides a straightforward derivation of Racinet's theorem using Ecalle's theory, simplifying the understanding of the double shuffle Lie algebra structure.

Findings

Racinet's theorem follows naturally from Ecalle's theory.

The paper clarifies the algebraic structure of formal multiple zeta values.

It introduces only the essential parts of Ecalle's theory for this derivation.

Abstract

The real multiple zeta values $\zeta(k_1,\ldots,k_r)$ are known to form a ${\bf Q}$-algebra; they satisfy a pair of well-known families of algebraic relations called the double shuffle relations. In order to study the algebraic properties of multiple zeta values, one can replace them by formal symbols $Z(k_1,\ldots,k_r)$ subject only to the double shuffle relations. These form a graded Hopf algebra over ${\bf Q}$, and quotienting this algebra by products, one obtains a vector space. A difficult theorem due to G. Racinet proves that this vector space carries the structure of a Lie coalgebra; in fact Racinet proved that the dual of this space is a Lie algebra, known as the double shuffle Lie algebra $ds$. J. Ecalle developed a deep theory to explore combinatorial and algebraic properties of the formal multiple zeta values. His theory is sketched out in some publications. However, because of the depth and complexity of the theory, Ecalle did not include proofs of many of the most important assertions, and indeed, even some interesting results are not always stated explicitly. The purpose of the present paper is to show how Racinet's theorem follows in a simple and natural way from Ecalle's theory.This necessitates an introduction to the theory itself, which we have pared down to only the strictly necessary notions and results.