Mould Theory and the Double Shuffle Lie Algebra Structure

Abstract

The real multiple zeta values ζ(k 1 , . . . , k r ) are known to form a 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 , . . . , k r ) subject only to the double shuffle relations. These form a graded Hopf algebra over 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 (essentially [E1] and [E2]). 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.

On Ecalle's and Brown's Polar Solutions to the Double Shuffle Equations Modulo Products

“…The Ihara bracket will be recalled in Section 4.1. For the proof of Theorem 1.2, there is another shorter exposition given by Brown (see Remark 4.7), but we repeat quickly Ecalle's theory of moulds [8] from [20,21], since it already involves Ecalle's construction of polar solutions. On this side, another Lie bracket, called the ari bracket { , } ari (see Section 3.2), comes into play.…”

“…On this side, another Lie bracket, called the ari bracket { , } ari (see Section 3.2), comes into play. We will first show an explicit connection with the Ihara bracket in Proposition 4.5 and then rephrase Theorem 7.2 of [20] in our notation. As a consequence, we obtain the explicit Lie isomorphism (see Section 5.1)…”

“…, x r ) (in particular, f (0) ∈ Q is a constant). In [20], these objects are called 'rational-function-valued moulds' and they are a special case of the moulds introduced by Ecalle [8]; a mould is a sequence of multivariable functions. For any such sequence, we will call f (r ) the depth-r component of f .…”

“…In this section, we briefly review Ecalle's theory from [8,20,21] with our notational conventions and translate it into our spaces.…”

“…We now define the ari bracket [8,20] translated into our setting. The order of composition is reversed from [20, equation (2.10)], for easier comparison with the Ihara bracket.…”