Mould Theory and the Double Shuffle Lie Algebra Structure

Abstract

The real multiple zeta values \(\zeta (k_1,\dots ,k_r)\) are known to form a \(\mathbb {Q}\)-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 \(\mathbb {Q}\), and quotienting this algebra by products, one obtains a vector space. A complicated 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 \(\mathfrak {ds}\). J. Ecalle developed a new 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.