Keywords:

Lie π 1 (M T M )→Lie π 1 (M EM ) construite par Hain and Matsumoto ([HM]), et avec la section grt→grt ell construite par Enriquez qui envoie l'algèbre de Lie grt de Grothendieck-Teichmüller vers sa version elliptique grt ell . Ces compatibilités sont exprimées par la commutativité du diagramme (A) (à l'exception de la flèche en pointillé, qui est conjecturale.)

Elliptic double shuffle, Grothendieck-Teichmüller and mould theory

Leila Schneps

Abstract. In this article we define an elliptic double shuffle Lie algebra ds ell that generalizes the well-known double shuffle Lie algebra ds to the elliptic situation. The double shuffle, or dimorphic, relations satisfied by elements of the Lie algebra ds express two families of algebraic relations between multiple zeta values that conjecturally generate all relations. In analogy with this, elements of the elliptic double shuffle Lie algebra ds ell are Lie polynomials having a dimorphic property called ∆-bialternality that conjecturally describes the (dual of the) set of algebraic relations between elliptic multiple zeta values, which arise as coefficients of a certain elliptic generating series (constructed explicitly in [LMS]) and closely related to the elliptic associator defined by Enriquez ([En1]). We show that one of Ecalle's major results in mould theory can be reinterpreted as yielding the existence of an injective Lie algebra morphism ds→ds ell . Our main result is the compatibility of this map with the tangential-base-point section Lie π 1 (M T M )→Lie π 1 (M EM ) constructed by Hain and Matsumoto and with the section grt→grt ell mapping the Grothendieck-Teichmüller Lie algebra grt into the elliptic Grothendieck-Teichmüller Lie algebra grt ell constructed by Enriquez. This compatibility is expressed by the commutativity of the following diagram (excluding the dotted arrow, which is conjectural).

Lie π 1 (M T M )

Brown / / Résumé. Dans cet article, nous définissons une algèbre de Lie de double mélange elliptique ds ell qui généralise l'algèbre de Lie bien connue de double mélange ds au cas elliptique. Les relations de double mélange (ou dimorphiques) satisfaites par les éléments de l'algèbre de Lie ds expriment deux familles de relations algébriques entre les valeurs zêta multiples, qui engendrent conjecturalement toutes les relations. En analogie avec cette conjecture, les éléments de l'algèbre de double mélange elliptique ds ell sont des polynômes de Lie ayant une propriété dimorphique, appelée ∆-bialternalité, qui décrit conjecturalement (le dual de) l'ensemble des relations algébriques entre les valeurs zêtas elliptiques multiples, qui sont les coefficients d'une certaine série génératrice elliptique (construite explicitement dans [LMS]) reliée à l'associateur d 'Enriquez ([En1]). Nous montrons que l'un des résultats majeurs de la théorie des moules de J. Écalle peut être interprété comme l'existence d'un morphisme injectif ds→ds ell d'algèbres de Lie. Notre résultat principal est la compatibilité de ce morphisme avec la section "point base tangentiel"

1. Introduction

Overview

The goal of this paper is to apply Ecalle's mould theory to define an elliptic double shuffle Lie algebra ds ell that turns out to parallel Enriquez' construction in [START_REF] Enriquez | Elliptic Associators[END_REF] of the elliptic Grothendieck-Teichmüller Lie algebra, and Hain and Matsumoto's construction of the fundamental Lie algebra of the category M EM of mixed elliptic motives in [HM]. Both of those Lie algebras are equipped with canonical surjections to the corresponding genus zero Lie algebras,

grt ell → → grt Lie π 1 (M EM ) → → Lie π 1 (M T M ).
Here, M T M is the category of mixed Tate motives over Z, and the notation Lie π 1 (M T M ) (resp. Lie π 1 (M EM )) denotes the Lie algebra of the pro-unipotent radical of the fundamental group of the Tannakian category M T M (resp. M EM ) equipped with the de Rham fiber functor (resp. its lift to a fiber functor on M EM via composition with the natural surjection M EM → M T M , cf. [START_REF] Hain | Universal mixed elliptic motives[END_REF]§5].)

Each of the Lie algebras grt ell and Lie π 1 (M EM ) is also equipped with a natural section of the above surjection, corresponding, geometrically, to the tangential base point at infinity on the moduli space of elliptic curves:

γ : grt → grt ell γ t : Lie π 1 (M T M ) → Lie π 1 (M EM ).
Hain-Matsumoto determine a canonical Lie ideal of u of Lie π 1 (M EM ), and Enriquez defines a canonical Lie ideal r ell of grt ell , such that the above sections give semi-direct product structures grt ell r ell γ(grt) Lie π 1 (M EM ) u γ t Lie π 1 (M T M ) . All these maps are compatible with the canonical injective morphism Lie π 1 (M T M ) → grt whose existence was proven by Goncharov and Brown in two stages as follows. Goncharov constructed a Hopf algebra A of motivic zeta values as motivic iterated integrals [G, §5], and identified it with a subalgebra of the Hopf algebra of framed mixed Tate motives [G, §8]; he showed that these motivic zeta values satisfy the associator relations. Brown [START_REF] Brown | Mixed Tate motives over Z[END_REF] subsequently lifted Goncharov's construction to an algebra H in which the motivic ζ m (2) is non-zero, such that in fact

H A ⊗ Q[ζ m (2)]
. He was able to compute the structure and the dimensions of the graded parts of H and thus of A, from which it follows that A is in fact equal to the full Hopf algebra of framed mixed Tate motives. In the dual situation, this means that the fundamental Lie algebra of M T M injects into the Lie algebra of associators, namely the top arrow of the following commutative diagram:

Lie

π 1 (M T M ) / / grt Lie π 1 (M EM ) / / grt ell .
The elliptic double shuffle Lie algebra ds ell that we define in this article is conjecturally isomorphic to Lie π 1 (M EM ) and grt ell . We show that it shares with them the following properties: firstly, it comes equipped with an injective Lie algebra morphism

γ s : ds → ds ell ,
where ds is the regularized double shuffle Lie algebra defined in [R], where it is denoted dmr ("double mélange régularisé").

Secondly there is an injective derivation representation

ds ell → Der 0 Lie[a, b].
Unfortunately, we have not yet been able to find a good canonical Lie ideal in ds ell that would play the role of u and r ell , although it is easy to show that there is an injection b 3 → ds ell whose image conjecturally plays this role (cf. the end of section 1.3). Since u → b 3 → ds ell , we do have a Lie algebra injection, Lie π 1 (M EM ) → ds ell , but not the desired injection grt ell → ds ell , (the dotted arrow in the diagram in the abstract), which would follow as a consequence of Enriquez' conjecture that r ell = b 3 . It would have been nice to give a direct proof of the existence of a Lie algebra morphism grt ell → ds ell even without proving Enriquez' conjecture, but we were not able to find one. This result appears like an elliptic version of Furusho's injection grt → ds (cf. [F]), and may possibly necessitate some similar techniques.

Our main result, however, is the commutation of the diagram given in the abstract, which does not actually require an injective map grt ell → ds ell , but, given all the observations above, comes down to the commutativity of the triangle diagram

grt / / % % K K K K K K K K K K ds
y y t t t t t t t t t t Der 0 Lie [a, b] .

(1.1.1)

The morphisms from grt and ds to Der Lie [a, b] factor through the respective elliptic Lie algebras (cf. the diagram in the abstract). Note that the morphisms in (1.1.1) must not be confused with the familiar Ihara-type morphism grt → Der Lie[x, y] via y → [ψ(-x -y, y), y] and x + y → 0, and the analogous map for ds investigated in [S2]. [a, b], and that in particular this is the case for the derivations in the image of grt and ds (cf. section 2). This gives a direct interpretation of the two maps to derivations in the diagram (1.1.1) whose commutativity we prove.

The existence of the injection ds → ds ell arose from an elliptic reinterpretation of a major theorem by Ecalle in mould theory. This reading of Ecalle's work and interpretation of some of his important results constitute one of the main goals of this paper in themselves. Indeed, it appears that Ecalle's seminal work in mould and multizeta theory has been largely ignored by the multiple zeta community 3 . This minimalist way of phrasing the main result shows that it could actually be stated and proved without even defining an elliptic double shuffle Lie algebra. However, this object is important in its own right, principally for the following reason. Recall that the usual double shuffle Lie algebra ds expresses the double shuffle relations satisfied by the multiple zeta values, in the following sense. Let FZ, the formal multizeta algebra, be the graded dual of the universal enveloping algebra of ds; it is generated by formal symbols satisfying only the double shuffle relations. Since motivic and real multizeta values are known to satisfy them (see for example [So]), FZ surjects onto the algebras of motivic and real multizeta values. These surjections are conjectured to be isomorphisms, i.e., it is conjectured that the double shuffle relations generate all algebraic relations between motivic, resp. real multizeta values (with the first of these problems being undoubtedly much more tractable than the second, for reasons of transcendence).

The role played by the double shuffle algebra with respect to ordinary multizeta values is analogous to the role played by the elliptic double shuffle algebra defined in this article with respect to the elliptic mzv's defined in [LMS]. There, we define an elliptic generating series in the completed Lie algebra Lie[a, b], whose coefficients, called elliptic mzv's or emzv's, are related to the iterated integrals that form the coefficients of Enriquez' monodromic elliptic associator, and we give an explicit "dimorphic" or "double shuffle" type symmetry of this generating series which is exactly the defining property of ds ell . Indeed, letting E denote the graded Hopf algebra generated 3 According to the author's discussion with several colleagues, this appears to be at least partly due to a reluctance to accept Ecalle's language, because, at least according to some, it uses a system of words with varying vowels, rather than the more standard single letters, for the basic objects. This seems surprising, as it is unclear why calling a derivation arit(f), say, rather than D f should pose such a problem. Possibly we enter here into the domain of psychology. A second, more serious obstacle is the lack of proofs in Ecalle's work, and the incredible profusion of statements, which makes it difficult to pick out exactly what is needed to establish a specific result. The author has attempted to solve this problem, at least partially, in the basic text [S] which gives an introduction with complete proofs to the portion of Ecalle's work most directly related to current problems in double shuffle algebra.

by the emzv's, we show in [LMS] that the vector space ne = E >0 / E >0 2 is isomorphic to a semidirect product b 3 nz ∨ , where nz is the space of "new multizeta values" obtained by quotienting the algebra of multizeta values by ζ(2) and products. Under the standard conjecture from multizeta theory nz ∨ grt, as well as Enriquez' conjecture r ell b 3 , this implies that ne grt ell . If grt ell ds ell , as we believe, this would mean that the elliptic double shuffle property determines all algebraic relations between the emzv's. This topic, which reflects the geometric aspects of the elliptic double shuffle relations introduced in this paper, is explored in detail in [LMS].

The content of the present paper has some relation with the recent preprint [START_REF] Brown | Zeta elements in depth 3 and the fundamental Lie algebra of a punctured elliptic curve[END_REF] as well as the earlier, closely related online lecture notes [START_REF] Brown | Talk on Anatomy of Associators[END_REF]. In particular Brown gives the existence of rational-function moulds satisfying the double shuffle relations, which is an immediate consequence of an important theorem of Ecalle that appears in all of his articles concerning ARI/GARI and multiple zeta values (cf. Theorem 1.3.2 below), although Brown introduces a completely different construction (vines and grapes). Brown also mentions in passing (cf. (3.7) of [START_REF] Brown | Zeta elements in depth 3 and the fundamental Lie algebra of a punctured elliptic curve[END_REF]) the result of the useful extension Lemma 2.1.2 below, however without proof. In [START_REF] Brown | Talk on Anatomy of Associators[END_REF] (conjecture 3) and [START_REF] Brown | Zeta elements in depth 3 and the fundamental Lie algebra of a punctured elliptic curve[END_REF] (following Prop. 4.6), Brown asks the question of whether u geom pls. The answer to this question is no; indeed all elements of grt with no depth 1 part furnish elements of pls not lying in u via Enriquez' section, as explained in the Corollary following Theorem 1.3.3.

Acknowledgements. The work on this paper benefited from discussions with B. Enriquez and P. Lochak, both of whom listened patiently and provided some crucial elements of proof. J. Ecalle repeatedly gave of his time to help understand some of his results. R. Hain also shed some light on details arising from his motivic work. I thank them all warmly.

The elliptic Grothendieck-Teichmüller Lie algebra

In this section we recall the definition of the elliptic Grothendieck-Teichmüller Lie algebra grt ell defined in [START_REF] Enriquez | Elliptic Associators[END_REF], along with some of its main properties. Recall that the genus 1 braid Lie algebra on n strands, t 1,n , is generated by elements x + 1 , . . . , x + n and x - 1 , . . . , x - n subject to relations

x + 1 + • • • + x + n = x - 1 + • • • + x - n = 0, [x + i , x + j ] = [x - i , x - j ] = 0 if i = j, [x + i , x - j ] = [x + j , x - i ] for i = j, [x + i , [x + j , x - k ]] = [x - i , [x + j , x - k ]] = 0 for i, j, k distinct. We write t ij = [x + i , x - j ].
It is the Lie algebra of the unipotent completion of the topological fundamental group of the configuration space of n ordered marked points on the torus (cf. [START_REF] Calaque | Universal KZB equations: the elliptic case, Algebra, Arithmetic and Geometry: in honor of Yu[END_REF]§2.2] for details). The Lie algebra t 1,2 is isomorphic to Lie[a, b], the free Lie algebra on two generators4 a and b. Throughout this article, we write Lie [a, b] for the completed Lie algebra, i.e., it contains infinite Lie series and not just polynomials. Thus an element α ∈ t 1,2

Lie[a, b] is a Lie series α(a, b) in two free variables.

Definition. The elliptic Grothendieck-Teichmüller Lie algebra grt ell is the set of triples (ψ, α

+ , α -) with ψ ∈ grt, α + , α -∈ t 1,2 , such that setting            Ψ(x ± 1 ) = α ± (x ± 1 , x ∓ 1 ) + [x ± 1 , ψ(t 12 , t 23 )], Ψ(x ± 2 ) = α ± (x ± 2 , x ∓ 2 ) + [x ± 2 , ψ(t 12 , t 13 )], Ψ(x ± 3 ) = α ± (x ± 3 , x ∓ 3 ) (1.2.1)
yields a derivation of t 1,3 . The space grt ell is made into a Lie algebra by bracketing derivations; in other words, writing

D α ± for the derivation of t 1,2 Lie[a, b] which takes a → α + (a, b) and b → α -(a, b), we have (ψ, α + , α -), (φ, β + , β -) = {ψ, φ}, D α ± (β + ) -D β ± (α + ), D α ± (β -) -D β ± (α -) ,
where {ψ, φ} is the Poisson (or Ihara) bracket on grt. Finally, we assume that the coefficient of a in both α + and α -is equal to 0.

Remark. The last assumption is not contained in Enriquez' original definition. In particular he allows the element (0, 0, a), corresponding to the derivation e(a) = 0, e(b) = a, which together with 0 (a) = b, 0 (b) = 0 generate a copy of sl 2 in grt ell . Because of this, Enriquez' version of grt ell is not pronilpotent, and is thus strictly larger than the Lie π 1 (M EM ) studied in [HM], which is the Lie algebra of the prounipotent radical of the fundamental group of M EM . Thus, isomorphism can only be conjectured if the extra element is removed, motivating our slight alteration of his definition. We nonetheless write grt ell for the modified version; the results of Enriquez on elements of grt ell that we cite adapt directly with no changes.

We summarize Enriquez' important results concerning grt ell in the following theorem.

Theorem 1.2.1. (cf. [START_REF] Enriquez | Elliptic Associators[END_REF]) For all (ψ, α + , α -) ∈ grt ell , the derivation D α ± of t 1,2 annihilates the element t 12 = [a, b]. But for each ψ ∈ grt, there exists one and only one triple (ψ, α + , α -) ∈ grt ell such that D α ± restricts to the Lie subalgebra Lie[t 01 , t 12 ] as follows:

t 01 → [ψ(t 01 , t 12 ), t 01 ], t 02 → [ψ(t 02 , t 12 ), t 02 ], t 12 → 0. (1.2.2)
The map γ : grt → grt ell mapping ψ to this triple is a Lie algebra morphism that is a section of the canonical surjection grt ell → grt. The Lie algebra grt ell thus has a semi-direct product structure grt ell = r ell γ(grt).

(1.2.3)

These results of Enriquez show that grt ell is generated by elements belonging to two particular subspaces. The first is the subspace r ell of triples (ψ, α + , α -) with ψ = 0, which forms a Lie ideal inside grt ell . The quotient grt ell /r ell is canonically isomorphic to grt, the surjection being nothing other than the morphism forgetting α + and α -. The second subspace, the image of the section grt → grt ell , is the space of triples that restrict on the free Lie subalgebra Lie[t 01 , t 02 ] to Ihara-type derivations (1.2.2). For any triple (ψ, α + , α -) of the second type, i.e., in the -but only and uniquely for those, not for general elements of grt ell -we let D ψ = D α ± , and write Dψ for the the restriction of D ψ to Lie[t 02 , t 12 ] given by (1.2.2).

Remark. This is actually a rephrasing of part of Enriquez' results. In fact, he gives the derivation D ψ by explicitly displaying its value on t 01 (as in (1.2.2) and on b. Since D ψ (t 12 ) = 0, the restriction of D ψ to Lie[t 01 , t 02 ] is the well-known Ihara derivation associated to ψ ∈ grt, and therefore the value on t 02 must be as in (1.2.2). The fact that D ψ is the only extension of (1.2.2) to a derivation on all of Lie[a, b] follows from our extension Lemma 2.1.2 below. This characterization of D ψ is sufficient for our purposes in this article; we do not actually use the explicit expression of D ψ (b), but it is necessary for Enriquez' work on elliptic associators.

The map

grt ell → Der 0 Lie[a, b] (ψ, α + , α -) → D α ±
is injective; in other words, knowing the pair (α + , α -) allows us to uniquely recover ψ. This result follows from [T2, Theorem 1.17] (building on previous work in [NTU]), which states that removing the third braid strand yields an injection D

(2) 1 → D

(1)

1 , where D

(1) 1

Der 0 Lie[a, b] and D

(2) 1 is a space of special derivations of L

(2) 1 t 1,3 which contains (and is conjecturally equal to) grt ell . Furthermore, by Lemma 2.1.1 below, there is an injective linear map

Der 0 Lie[a, b] → Lie[a, b] D → D(a), (1.2.4)
which is a Lie algebra bijection onto its image when that image (equal to the subspace Lie push [a, b] of push-invariant elements of Lie[a, b], cf. section 2) is equipped with the corresponding bracket.

In particular this shows that in the triple (ψ, α + , α -), the element α + determines α -, and thus also ψ. We write γ + : grt → Lie[a, b] for the map sending ψ → α + . By the above arguments, γ + determines γ and vice versa.

The desired triangle diagram (1.1.1) is thus equivalent to

grt γ + % % K K K K K K K K K K / / ds y y t t t t t t t t t t Lie push [a, b],
(1.2.5) by composing it with the map (1.2.4). Our main result, Theorem 1.3.1 below, is the explicit version of the commutation of the diagram (1.2.5).

Mould theory, elliptic double shuffle and the main theorem

In this section we explain how we use Ecalle's mould theory -particularly adapted to the study of dimorphic (or "double shuffle") structures -to construct the elliptic double shuffle Lie algebra ds ell , which like grt ell is a subspace of the push-invariant elements of Lie [a, b], and how we reinterpret one of Ecalle's major theorems and combine it with some results from Baumard's Ph.D. thesis ( [B]), to define the injective Lie morphism ds → ds ell .

We assume some familiarity with moulds in this section; however all the necessary notation and definitions starting with that of a mould are recalled in the appendix at the end of the paper. We use the notation ARI to denote the vector space of moulds with constant term 0, and write ARI lu for ARI equipped with the lu-bracket and ARI ari for ARI equipped with the ari-bracket (the usual ARI according to Ecalle's notation). Similarly, we write GARI for the set of moulds with constant term 1 and write GARI mu and GARI gari for the groups obtained by equipping GARI with the mu and gari multiplication laws. In section 3 we will introduce a third Lie bracket on ARI, the Dari-bracket, and employ the notation ARI Dari , as well as the corresponding group GARI Dgari with multiplication law Dgari.

We define the following operators on moulds:

     dar(P )(u 1 , . . . , u r ) = u 1 • • • u r P (u 1 , . . . , u r ) dur(P )(u 1 , . . . , u r ) = (u 1 + • • • + u r ) P (u 1 , . . . , u r ) ∆(P )(u 1 , . . . , u r ) = u 1 • • • u r (u 1 + • • • + u r ) P (u 1 , . . . , u r ) ad(Q) • P = [Q, P ] for all Q ∈ ARI. (1.3.1)
We take dar(P )(∅) = dur(P )(∅) = ∆(P )(∅) = P (∅). The operators dur and ad(Q) are derivations of the Lie algebra ARI lu , whereas dar is an automorphism of ARI lu . We will also make use of the inverse operators dur -1 (resp. dur -1 and ∆ -1 ) defined by dividing a mould in depth r by (u

1 + • • • + u r ) (resp. by (u 1 + • • • + u r ) and (u 1 + • • • + u r )u 1 • • • u r ). If p ∈ Lie[a, b], then we have    ma [p, a] = dur ma(p) ma p(a, [b, a]) = dar ma(p) ma [p(a, [b, a]), a] = ∆ ma(p) . (1.3.2)
A proof of the first equality can be found in [START_REF] Racinet | Séries génératrices non commutatives de polyzêtas et associateurs de Drinfel[END_REF]Proposition 4.2.1.1] or [START_REF] Schneps | An Introduction to Ecalle's theory of moulds[END_REF]Lemma 3.3.1]. The second is obvious from the definition of ma (cf. Appendix), since substituting [b, a] for b in C k yields -C k+1 so making the substitution in a monomial

C k 1 • • • C k r yields (-1) r C k 1 +1 • • • C k r +1 , and we have ma (-1) r C k 1 +1 • • • C k r +1 = (-1) r (-1) k 1 +•••+k r u k 1 1 • • • u k r r = u 1 • • • u r ma C k 1 • • • C k r .
The third equality of (1.3.2) follows from the first two.

We now recall the definition of the key mould pal that lies at the heart of much of Ecalle's theory of moulds. Following [E2], we start by introducing an auxiliary mould dupal ∈ ARI, given by the simple explicit expression

dupal(u 1 , . . . , u r ) = B r r! 1 u 1 • • • u r r-1 j=0 (-1) j r -1 j u j+1 . (1.3.3)
The mould pal is then defined by setting pal(∅) = 1 and using the equality

dur(pal) = pal dupal, (1.3.4)
which gives a recursive definition for pal depth by depth starting with pal(∅) = 1, since to determine the left-hand side dur(pal) in depth r only requires knowing pal up to depth r -1 on the right-hand side.

Since pal(∅) = 1, we have pal ∈ GARI. We write invpal for its inverse inv gari (pal) in the group GARI gari . Since GARI gari is the exponential of the Lie algebra ARI ari , it has an adjoint action on ARI ari ; we write Ad ari (P ) for the adjoint operator on ARI ari associated to a mould P ∈ GARI gari .

At this point we are already equipped to baldly state our main theorem linking Ecalle's theory of moulds to Enriquez' section γ : grt → grt ell , or rather to the modified version γ + introduced above that maps ψ to the associated element α + in Enriquez' triple (ψ, α + , α -).

Theorem 1.3.1. Let ψ ∈ grt and set f (x, y) = ψ(x, -y). We have the following equality of moulds:

∆ Ad ari (invpal) • ma(f ) = ma γ + (ψ) . (1.3.5)
In order to place this theorem in context and explain its power in terms of helping to define an elliptic double shuffle Lie algebra that in turn will shed light on the dimorphic ("double-shuffle") properties of elliptic multiple zeta values, we first give some results from the literature, starting with Ecalle's main theorem, with which he first revealed the surprising role of the adjoint operator Ad ari (pal) and its inverse Ad ari (pal) -1 = Ad ari (invpal).

Recall from the appendix that in terms of moulds, ds is isomorphic to the Lie subalgebra of ARI ari of polynomial-valued moulds that are even in depth 1, and are alternal with swap that is alternil up to addition of a constant mould. The notation we use for this in mould language is a bit heavy, but has the advantage of concision and total precision in that the various symbols attached to ARI carry all of the information about the moulds in the subspace under consideration: we have the isomorphism ma :

ds ma → ARI pol,al * il ari
, where pol indicates polynomial moulds, the underlining is Ecalle's notation for moulds that are even in depth 1, and the usual notation al/il for an alternal mould with alternil swap is weakened to al * il when the swap is only alternil up to addition of a constant mould.

Similarly, the notation ARI al * al ari refers to the subspace of moulds in ARI ari that are even in depth 1 and alternal with swap that is alternal up to addition of a constant mould (or "bialternal"). When we consider the subspace of these moulds that are also polynomial-valued, ARI pol,al * al , we obtain the (image under ma of the) "linearized double shuffle" space ls studied for example in [START_REF] Brown | Zeta elements in depth 3 and the fundamental Lie algebra of a punctured elliptic curve[END_REF]. But the full non-polynomial space is of course hugely larger. One of Ecalle's most remarkable discoveries is that the mould pal provides an isomorphism between the two types of dimorphy, as per the following theorem. One important point to note in the result of Theorem 1.3.2 is that the operator Ad ari (invpal) does not respect polynomiality of moulds. Indeed, applying Ad ari (pal) to bialternal polynomial moulds produces quite complicated denominators with many factors. However, in his doctoral thesis S. Baumard was able to show that conversely, when applying Ad ari (invpal) to moulds ma(f ) for f ∈ ds, i.e., to moulds in ARI pol,al * il , the denominators remain controlled. Indeed, let ARI ∆ denote the space of moulds P ∈ ARI such that ∆(P ) ∈ ARI pol , i.e., the space of rational-function valued moulds whose denominator is "at worst"

u 1 • • • u r (u 1 + • • • + u r ) in depth r.
Theorem 1.3.3. [START_REF] Baumard | Aspects modulaires et elliptiques des relations entre multizêtas[END_REF]Thms. 3.3,4.35] The space ARI ∆ forms a Lie algebra under the ari-bracket, and we have an injective Lie algebra morphism

Ad ari (invpal) : ARI pol,al * il ari → ARI ∆ ari . (1.3.7)
Recall that pls ("polar linearized double shuffle") is the notation used by F. Brown for the space ARI ∆,al/al and u for the Lie subalgebra of ARI generated by B -2 and B 2i for i ≥ 1, where B i denotes the mould concentrated in depth 1 defined by B i (u 1 ) = u i 1 . As a corollary of Theorems 1.3.1, 1.3.2 and 1.3.3, we give a negative answer to the question posed by Brown ([Br2], conjecture 3 and [START_REF] Brown | Zeta elements in depth 3 and the fundamental Lie algebra of a punctured elliptic curve[END_REF], following Prop. 4.6) as to whether pls and u are equal.

Corollary. Let ψ ∈ grt be an element of grt having no depth 1 part. Then

∆ -1 ma(γ + (ψ) ∈ ARI ∆,al/al = pls but ∆ -1 ma(γ + (ψ)) / ∈ u.
Proof. Since by Furusho's theorem, ψ(x, y) → f (x, y) = ψ(x, -y) maps grt → ds, we have ma(f ) ∈ ARI pol,al * il for every ψ ∈ grt. In particular, if ψ has no depth 1 part, then ma(f ) ∈ ARI pol,al/il ; thus by Theorem 1.3.2, Ad ari (invpal) • ma(f ) ∈ ARI al/al , and by Theorem 1.3.3, it also lies in ARI ∆ ; thus it lies in ARI ∆,al/al = pls. By Theorem 1.3.1, Ad ari (invpal) • ma(f ) is equal to ∆ -1 ma(γ + (ψ) where γ + denotes Enriquez' section grt → grt ell , associating to ψ ∈ grt the element α + from the triple (ψ, α + , α -). But Enriquez shows that grt ell is a semi-direct product γ + (grt) r ell and that ∆(u) ⊂ ma(r ell ). Thus ma γ + (grt) ∩ ∆(u) = {0}. ♦

For the rest of this article we will use the notation:

     f = ψ(x, -y) F = ma(f ) A = Ad ari (invpal) • F M = ∆(A).
(1.3.8)

Corollary 1.3.4. Let f ∈ ds and let F = ma(f ), so F ∈ ARI pol,al * il . Then the mould M = ∆ Ad ari (invpal) • F is alternal, push-invariant and polynomial-valued.

Proof. Let A = Ad ari (invpal) • F . Then A ∈ ARI al * al by Theorem 1.3.2, so A is alternal, and furthermore A is push-invariant because all moulds in ARI al * al are push-invariant (see [E2] or [START_REF] Schneps | An Introduction to Ecalle's theory of moulds[END_REF]Lemma 2.5.5]). Thus M = ∆(A) is also alternal and push-invariant since ∆ preserves these properties. The fact that M is polynomial-valued follows from Theorem 1.3. , so the image under ∆ • Ad ari (invpal) lies in the space of polynomial-valued ∆-bialternal moulds that are also even in depth 1 (since it is easy to see that Ad ari (invpal) preserves the lowest-depth part of a mould). Thus we can define γ s to be the polynomial avatar of ∆ • Ad ari (invpal), i.e., γ s is defined by the commutation of the diagram

ds ma / / γ s ARI pol,al * il ari ∆•Ad ari (invpal) ds ell ma / / ∆ ARI ∆,al * al ari
.

(1.3.10)

Thus for f ∈ ds we have

ma γ s (f ) = ∆ Ad ari (invpal) • ma(f ) .
This reduces the statement of the main Theorem 1.3.1 above to the equality

γ s (f ) = γ + (ψ),
i.e., to the commutation of the diagram

grt / / γ + % % K K K K K K K K K K ds γ s y y t t t t t t t t t t Lie push [a, b],
which is the precise version of the desired diagram (1.2.5).

As a final observation, we note that the definition of ds ell makes the injective Lie algebra morphism b 3 → ds ell mentioned at the beginning of the introduction obvious. Indeed, identifying b 3 with its image in Lie push [a, b] under the map (1.2.4), it is generated by the polynomials 2i (a) = ad(a) 2i (b) = C 2i+1 for i ≥ 0, which map under ma to the moulds B 2i concentrated in depth 1 and given by B 2i (u 1 ) = u 2i 1 (Ecalle denotes these moulds by ekma 2i at least for i ≥ 1; note however that B 0 and ∆ -1 (B 0 ) = B -2 are essential in the elliptic situation). To show that these moulds lie in ds ell , we need only note that the moulds ∆ -1 (B 2i ) = B 2i-2 are even in depth 1, and trivially bialternal since this condition is empty in depth 1.

Proof of the main theorem

For the proof of the main theorem, we first recall in 2.1 a few well-established facts about non-commutative polynomials, moulds and derivations, and give the key lemma about extending derivations on the Lie subalgebra Lie[t 01 , t 02 ] to all of Lie [a, b]. Once these ingredients are in place, the proof of the main theorem, given in 2.2, is a simple consequence of one important proposition, whose proof, contained in section 3, necessitates some developments in mould theory. In fact, the present section could be written entirely in terms of polynomials in a and b without any reference to moulds. We only use moulds in the proof of Lemma 2.1.1, but merely as a convenience, as even this result could be stated and proved in terms of polynomials. Indeed this has already been done (cf. [S2]), but the proof given here using moulds is actually more elegant and simple.

p = i≥0 (-1) i-1 i! a i b∂ i a (p a ) (2.1.1)
where ∂ a (a) = 1, ∂ a (b) = 0. We call p the partner of p. If P ∈ ARI then we define P to be the mould partner of P , given by the formula

P (u 1 , . . . , u r ) = 1 u 1 + • • • + u r P (u 2 , . . . , u r-1 , -u 1 -• • • -u r-1 ) -P (u 2 , . . . , u r ) . (2.1.2)
This formula defines a partner for any mould P ∈ ARI, but in the case of polynomial-valued moulds it corresponds to (2.1.1) in the sense that if P = ma(p), then P = ma(p ).

Recall that the push-operator on a mould is an operator of order r + 1 in depth r defined by

push(P )(u 1 , . . . , u r ) = P (-u 1 -• • • -u r , u 1 , . . . , u r-1 ),
and that a mould P is said to be push-invariant if P = push(P ). We say that a polynomial Let us first assume that P is push-invariant and P is its partner as given in (2.1.2). We have [P, B](u 1 , . . . , u r ) = P (u 1 , . . . , u r-1 ) -P (u 2 , . . . , u r ) (2.1.4) and dur(P

p ∈ Lie[a, b] is push-invariant if ma(p) is. Lemma 2.1.1. Let p,
) = P (u 2 , . . . , u r ) -P (u 2 , . . . , u r-1 , -u 1 -• • • -u r-1 ).
(2.1.5)

Thus [P, B] -dur(P ) is given in depth r > 1 by [P, B] = dur(P ), i.e.,

P (u 1 , . . . , u r-1 ) -P (u 2 , . . . , u r-1 , -u 1 -• • • -u r-1 ) = P -push -1 (P ) (u 1 , . . . , u r ), but since P is push-invariant, this is equal to zero, so by (2.1.3) D([a, b]) = 0. Assume now that D([a, b]) = 0, i.e.,
P (u 1 , . . . , u r-1 ) -P (u 2 , . . . , u r ) = (u 1 + • • • + u r )P (u 1 , . . . , u r ). (2.1.6)
This actually functions as a defining equation for P . But knowing that P = ma(p ) is a polynomial-valued mould, (2.1.6) implies that P (u 1 , . . . , u r-1 ) -P (u 2 , . . . , u r ) must vanish along the pole u 1 + • • • + u r = 0, in other words when u r = -u 1 -• • • -u r-1 , so we have

P (u 1 , . . . , u r-1 ) = P (u 2 , . . . , u r-1 , -u 1 -• • • -u r-1 ). (2.1.7)
As noted above, the right-hand side of (2.1.7) is nothing other than push -1 (P ), so (2.1.7) shows that P is push-invariant. Furthermore, we can substitute (2.1.7) into the left-hand side of (2.1.6) to find the new defining equation for P : Using this, we can easily prove the main theorem. Since t 12 = [a, b], we have E ψ (t 12 ) = 0, so Proposition 2.2.1 shows that E ψ restricts to a derivation Ẽψ on the Lie subalgebra Lie[t 02 , t 12 ], where it coincides with the restriction Dψ of Enriquez' derivation D ψ given in (1.2.2). Furthermore, since E ψ (t 12 ) = 0 and E ψ (a) = m is push-invariant, we are in the situation of Lemma 2.1.2 (ii), so

P (u 1 , . . . , u r ) = 1 u 1 + • • • + u r P (u 2 , . . . , u r-1 , -u 1 -• • • -u r-1 ) -P (u 2 , . . . , u r ) , ( 2

Mould theoretic derivations

We begin by defining a mould-theoretic derivation E ψ on ARI lu for each ψ ∈ grt as follows.

Definition. For any mould P , let Darit(P ) be the operator on moulds defined by

Darit(P ) = -dar arit ∆ -1 (P ) -ad ∆ -1 (P ) • dar -1 . (3.1.1)
Then for all P , Darit(P ) is a derivation of ARI lu , since arit(P ) and ad(P ) are both derivations and dar is an automorphism. Let ψ ∈ grt. We use the notation of (1.3.8), and set

E ψ = Darit(M ). (3.1.2)
Recall that ARI denotes the vector space of rational-valued moulds with constant term 0. Let ARI a denote the vector space obtained by adding a single generator a to the vector space ARI, and let ARI a lu be the Lie algebra formed by extending the lu-bracket to ARI a via the relation

[Q, a] = dur(Q) (3.1.3)
for every Q ∈ ARI lu . Recall from (1.3.2) that this equality holds in the polynomial sense if Q is a polynomial-valued mould; in other words, (1.3.3) extends to an injective Lie algebra morphism ma : Lie[a, b] → ARI a lu by formally setting ma(a) = a. The Lie algebra ARI lu forms a Lie ideal of ARI a lu , i.e., there is an exact sequence of Lie algebras 0 → ARI lu → ARI a lu → Qa → 0. We say that a derivation (resp. automorphism) of ARI lu extends to a if there is a derivation (resp. automorphism) of ARI a lu that restricts to the given one on the Lie subalgebra ARI lu . To check whether a given derivation (resp. automorphism) extends to a, it suffices to check that relation (3.1.3) is respected.

Recall that B = ma(b) is the mould concentrated in depth 1 given by B(u 1 ) = 1. Let us write B i , i ≥ 0, for the mould concentrated in depth 1 given by B i (u 1 ) = u i 1 . In particular (iii) For all P ∈ ARI, the derivation arit(P ) of ARI lu extends to a, taking the value arit(P )•a = 0. (iv) For all P ∈ ARI, the derivation Darit(P ) of ARI lu extends to a, with Darit(P ) • a = P . Furthermore, Darit(P ) • B 1 = 0.

B 0 = B = ma(b), and B 1 (u 1 ) = u 1 , so B 1 = ma([b, a]).
Proof. Since dar is an automorphism, to check (3.1.3) we write

[dar(Q), dar(a)] = [dar(Q), a] = dur dar(Q) .
But it is obvious from their definitions that dur and dar commute, so this is indeed equal to dar dur(Q) . This proves (i). We check (3.1.3) for (ii) similarly. Because dur(a) = 0 and dur is a derivation, we have

dur([Q, a]) = [dur(Q), a] = dur dur(Q) .
For (iii), we have

arit(P ) • [Q, a] = [arit(P ) • Q, a] = dur arit(P ) • Q) .
But as pointed out by Ecalle [E2] (cf. [S, Lemma 4.2.2] for details), arit(P ) commutes with dur for all P , which proves the result. For (iv), the calculation to check that (3.1.3) is respected is a little more complicated. Let Q ∈ ARI. Again using the commutation of arit(P ) with dur, as well as that of dar and dur, we compute

Darit(P ) • [Q, a] = Darit(P )(Q), a + Q, Darit(P )(a) = dur Darit(P ) • Q + [Q, P ] = -dur dar arit ∆ -1 (P ) • dar -1 (Q) -∆ -1 (P ), dar -1 (Q) + [Q, P ] = -dur dar arit ∆ -1 (P ) • dar -1 (Q) -dur Q, dur -1 (P ) + [Q, P ] = -dar dur arit ∆ -1 (P ) • dar -1 (Q) -[Q, N ], a + Q, [N, a]
with N = dur -1 P, i.e., P = [N, a]

= -dar arit ∆ -1 (P ) • dur dar -1 (Q) -[Q, a], N by Jacobi = -dar arit ∆ -1 (P ) • dar -1 dur(Q) -dur(Q), dur -1 P = -dar arit ∆ -1 (P ) • dar -1 dur(Q) -dar dar -1 dur(Q), dar -1 dur -1 (P ) = -dar arit ∆ -1 (P ) • dar -1 dur(Q) + dar ∆ -1 (P ), dar -1 dur(Q) = Darit(P ) • dur(Q).
This proves the first statement of (iv). For the second statement, we note that dar -1 (B 1 ) = B. Set R = ∆ -1 (P ). We compute

Darit(P ) • B 1 = -dar arit(R) • B + dar [R, B] = -u 1 • • • u r R(u 1 , . . . , u r-1 ) -R(u 2 , . . . , u r ) -u 1 • • • u r R(u 2 , . . . , u r ) -R(u 1 , . . . , u r-1 ) = 0.
Proof. Since M is push-invariant by Corollary 1.3.4, Darit(M ) preserves (ARI a lu ) pol,al by Lemma 3.1.3. Thus we are in the situation of Lemma 3.1.2, so there exists a derivation E M of Lie [a, b] such that E M (a) = m with ma(m) = M . Furthermore, setting B 1 = ma([b, a]), we know that Darit(M ) • B 1 = 0 by Lemma 3.1.1 (iv), and therefore by Lemma 3.1.2, we have This result means that we can now use mould theoretic methods to study Darit(M ) in order to prove Proposition 2.2.1.

E M ([b, a]) = E M ([a, b]) = 0.

The ∆-operator

Let us define a new Lie bracket, the Dari-bracket, on ARI by

Dari(P, Q) = Darit(P ) • Q -Darit(Q) • P,
where Darit(P ) is the lu-derivation defined in (3.1.1). Let ARI Dari denote the Lie algebra obtained by equipping ARI with this Lie bracket.

Proposition 3.2.1. The operator ∆ is a Lie algebra isomorphism from ARI ari to ARI Dari .

Proof. Certainly ∆ is a vector space isomorphism from ARI ari to ARI Dari since it is an invertible operator on moulds. To prove that it is a Lie algebra isomorphism, we need to show the Lie bracket identity ∆ ari(P, Q) = Dari ∆P, ∆Q , or equivalently,

Dari(P, Q) = ∆ ari(∆ -1 P, ∆ -1 Q) (3.2.1)
for all moulds P, Q ∈ ARI. But indeed, we have (3.2.4) Lemma 3.2.2. For any mould P ∈ GARI, the automorphism Dgarit(P ) of ARI lu extends to an automorphism of the Lie algebra ARI a lu with the following properties: i) its value on a is given by Dgarit(P ) • a = a -1 + P ∈ ARI a ;

Dari(P, Q) = Darit(P ) • Q -Darit(Q) • P = -dar • arit(∆ -1 P ) • dar -1 • Q + dar • ad(∆ -1 P ) • dar -1 • Q + dar • arit(∆ -1 Q) • dar -1 • P -dar • ad(∆ -1 Q) • dar -1 • P = -∆ • arit(∆ -1 P ) • ∆ -1 • Q + ∆ • arit(∆ -1 Q) • ∆ -1 • P + dar • ad(∆ -1 P ) • dar -1 • Q -dar • ad(∆ -1 Q) • dar -1 • P = -∆ • arit(∆ -1 P ) • ∆ -1 • Q + ∆ • arit(∆ -1 Q) • ∆ -1 • P + dar [∆ -1 (P ), dar -1 Q] -dar [∆ -1 (P ), dar -1 P ] = ∆ -arit(∆ -1 P • ∆ -1 Q + arit(∆ -1 Q) • ∆ -1 P + dur -1 [∆ -1 P, dar -1 Q] + [dar -1 P, ∆ -1 Q] = ∆ -arit(∆ -1 P • ∆ -1 Q + arit(∆ -1 Q) • ∆ -1 P + dur -1 [∆ -1 P, dur∆ -1 Q] + [dur∆ -1 P, ∆ -1 Q] = ∆ -arit(∆ -1 P • ∆ -1 Q + arit(∆ -1 Q) • ∆ -1 P + dur -1 dur [∆ -1 P, ∆ -1 Q] = ∆ -arit(∆ -1 P • ∆ -1 Q + arit(∆ -1 Q) • ∆ -1 P + [∆ -1 P, ∆ -1 Q] = ∆ ari(∆ -1 P, ∆ -1 Q) ,
(3.2.5)

ii) we have Dgarit(P )

• B 1 = B 1 .
Proof. Let Q = log Dari (P ) ∈ ARI. We saw in Lemma 3.1.1 (iv) that Darit(Q) extends to ARI a lu with Darit(Q) • a = Q. By diagram (3.2.4), we have

Dgarit(P ) • a = Dgarit exp Dari (Q) • a = exp Darit(Q) • a = a + Darit(Q) • a + 1 2 Darit(Q) 2 • a + • • • = a + Q + 1 2 Darit(Q) • Q + • • • = a -1 + exp Dari (Q) by (3.2.2) = a -1 + P.
The second statement follows immediately from the fact that Darit(Q) The main point is the following result decomposing Darit(M ) into three factors; a derivation conjugated by an automorphism. We note that although the values of the derivation and the automorphism in Proposition 3.3.1 on a are polynomial-valued moulds, this is false for their values on B = ma(b), which means that this decomposition is a result which cannot be stated in the power-series situation of Lie [a, b]; the framework of mould theory admitting denominators is crucial here.

Proposition 3.3.1. We have the following identity of derivations:

Darit ∆ Ad ari (invpal) • F = Dgarit ∆ * (invpal) • Darit ∆(F ) • Dgarit ∆ * (invpal) -1 . (3.3.1)
Proof. We use two standard facts about Lie algebras and their exponentials. Firstly, for any exponential morphism exp : g → G mapping a Lie algebra to its associated group, the natural adjoint action of G on g, denoted Ad g (exp(g)) • h, satisfies

exp Ad g exp(g) • h = Ad G exp(g) exp(h) = exp(g) * G exp(h) * G exp(g) -1 , (3.3.2)
where * G denotes the multiplication in G, defined by

exp(g) * G exp(h) = exp ch g (g, h) (3.3.3)
where ch g denotes the Campbell-Hausdorff law on g. Secondly, if ∆ : g → h is an isomorphism of Lie algebras, then the following diagram commutes:

g ∆ / / Ad g exp g (g) h Ad h exp h ∆(g) g ∆ / / h. (3.3.4)
To prove (3.3.1), we start by taking the exponential of both sides. Let lipal = log ari (invpal). We start with the left-hand side and compute

exp Darit ∆ Ad ari (invpal) • F = exp Darit ∆ Ad ari (exp ari (lipal)) • F = exp Darit Ad Dari exp Dari (∆lipal) • ∆(F ) = Dgarit exp Dari Ad Dari exp Dari (∆lipal) • ∆(F ) (3.3.5) = Dgarit exp Dari ∆lipal • Dgarit exp Dari ∆(F ) • Dgarit exp Dari ∆lipal -1 = Dgarit ∆ * (invpal) • exp Darit ∆(F ) • Dgarit ∆ * (invpal) -1 ,
where the second equality follows from (3.3.4) (with g, exp g and Ad g identified with ARI ari , exp ari and Ad ari , and the same three terms for h with the corresponding terms for ARI Dari ), the third from (3.2.4), the fourth from (3.3.2) and the fifth again from (3.2.4). But the first and last expressions in (3.3.5) are equal to the exponentials of the left-and right-hand sides of (3.3.1). This concludes the proof of the Proposition. ♦

We can now complete the proof of Proposition 2.2.1 by using Proposition 3.3.1 to compute the value of E ψ (t 02 ). By (3.2.9) and the Corollary to Proposition 3.2.3, we have

Dgarit ∆ * (invpal) • a = a -1 + ∆ * (invpal) = ma Ber -b (a) = ma(t 02 ). (3.3.6)
Recall that E ψ is nothing but the polynomial version of Darit(M ) restricted to the Lie algebra generated by the moulds a and B. Thus, to compute the value of E ψ on t 02 = Ber -b (a), we can now simply use (3.3.1) to compute the value of Darit(M ) on ma(t 02 ). By (3.3.6), the rightmost map of the right-hand side of (3.3.1) maps ma(t 02 ) to a. By Lemma 3.1.1 (iv), the derivation Darit(P ) for any mould P ∈ ARI extends to a taking the value P on a, so we can apply the middle map of (3.3.1) to a, obtaining

Darit ∆(F ) • a = ∆(F ) = dur dar(F ) = ma [f (a, [b, a]), a] = ma [ψ(a, [a, b]), a] = ma [ψ(a, t 12 ), a] . (3.3.7)
Finally, we note that by Lemma 3.2.2 (ii), the leftmost map of the right-hand side of (3.3.1) fixes B 1 = -ma(t 12 ), so it also fixes ma(t 12 ). By (3.3.6), it sends a to ma(t 02 ), so applying it to the rightmost term of (3.3.7) we obtain the total expression

Darit(M ) ma(t 02 ) = ma [ψ(t 02 , t 12 ), t 02 ] .
In terms of polynomials, this gives the desired expression

E ψ (t 02 ) = [ψ(t 02 , t 12 ), t 02 ],
which concludes the proof. ♦

Power series, moulds, standard multiplication and Lie bracket. Via the map (A.2), many of the familiar notions associated with power series and Lie series pass to polynomial moulds, with general expressions that are in fact valid for all moulds.

In particular, the standard mould multiplication mu is given by

mu(P, Q)(u 1 , . . . , u r ) = r i=0 P (u 1 , . . . , u i )Q(u i+1 , . . . , u r ).
For simplicity, we write P Q = mu(P, Q). The multiplication mu generalizes ordinary multiplication of non-commutative power series in the sense that

ma(f g) = mu ma(f ), ma(g) = ma(f ) ma(g) (A.4) for f, g ∈ Q C .
The space (M oulds) pol is a ring under the mu multiplication, generated by the depth 1 polynomial moulds B i given by B i (u 1 ) = u i 1 for i ≥ 0. By (A.4), the linear map ma from (A.2) can be defined as a ring isomorphism from Q C to (M oulds) pol , taking values ma(C i ) = (-1) i-1 B i-1 on the generators C i for i ≥ 1.

A mould P is invertible for the mu-multiplication if and only if its constant term P (∅) ∈ Q is invertible. If the constant term is 1, the formula for the mu-inverse P -1 = invmu(P ) is explicitly given by

P -1 (u) = 0≤s≤r (-1) s u=u 1 •••u s P (u 1 ) • • • P (u s ),
where the sum runs over all ways u 1 • • • u s of cutting the word u = (u 1 , . . . , u r ) into s nonempty chunks. By (A.4), if f ∈ Q C is invertible (i.e., has non-zero constant term), we have ma(f -1 ) = P -1 . The mu-multiplication makes GARI, the set of moulds with constant term 1, into a group that we denote by GARI mu . Defining the associated lu-bracket by lu(P, Q) = mu(P, Q) -mu(Q, P ), i.e., [P, Q] Let the swap operator on moulds be defined by swap(A)(v 1 , . . . , v r ) = A(v r , v r-1 -v r , . . . , v 1 -v 2 ).

Here the use of the alphabet v 1 , v 2 , . . . instead of u 1 , . . . , u r is purely a convenient way to distinguish a mould from its swap. The mould swap(A) is alternal if it satisfies the property (A.5) in the v i 's.

The space of moulds that are alternal and have a swap that is also alternal is denoted ARI al/al ; these moulds are said to be strictly bialternal. We particularly consider the situation where a mould is alternal and its swap differs from an alternal mould by addition of a constant-valued mould. Such moulds are called bialternal, and the space of bialternal moulds is denoted ARI al * al . The space of polynomial-valued bialternal moulds is denoted ARI pol,al * al . Finally, we recall that Ecalle uses the notation of underlining the symmetry of a mould to indicate that its depth 1 part is an even function of u 1 ; thus we use the notation ARI pol,al * al etc. to denote the subspaces of moulds that are even in depth 1. The subspace ARI pol,al * al ari forms a Lie algebra under the ari-bracket (cf. [START_REF] Schneps | An Introduction to Ecalle's theory of moulds[END_REF]Theorem 2.5.6]), which is isomorphic under the map ma to the "linearized double shuffle" Lie algebra ls studied for example in [START_REF] Brown | Zeta elements in depth 3 and the fundamental Lie algebra of a punctured elliptic curve[END_REF].

Ecalle introduces a second symmetry called alternility on moulds in the v i 's, which generalizes the usual stuffle relations on polynomials in a and b. As above, we write ARI al/il , ARI al * il and ARI al * il for the space of alternal moulds with swap that is alternil, resp. alternil up to addition of a constant mould, resp. also even in depth 1. The space ARI pol,al * il is isomorphic under the map ma to the double shuffle Lie algebra ds. [S, ??] Twisted Magnus automorphism and group law. Let G ⊂ Q C denote the set of power series with constant term 1, so that ma gives a bijection G → GARI pol to the set of polynomialvalued moulds with constant term 1. We write G for the group obtained by putting the standard power series multiplication on G, so that we have a group isomorphism G GARI pol mu . For all p ∈ G, we define the associated "twisted Magnus" automorphism A p of G, defined by A p (a) = a, A p (b) = pbp -1 . These automorphisms satisfy the composition law (A q • A p )(b) = A q (p)qbq -1 A(p) -1 , which defines a different multiplication on the set G, given by p q = A q (p)q = p a, qbq -1 q(a, b), (A.6) satisfying A p q = A q • A p .

The inverse of the automorphism A p is given by A q where q is the unique power series such that the right-hand side of (A.6) is equal to 1. We write G for the "twisted Magnus" group obtained by putting the multiplication law (A.6) on G. The association p → A p extends to the general case of moulds by associating to every P ∈ GARI the automorphism of GARI mu defined by Ecalle and denoted garit(P ), whose action on Q ∈ GARI is given by

garit(P ) • Q (u) = s≥0 u=a 1 b 1 c 1 •••a s b s c s Q( b 1 • • • b 2 )P (a 1 ) • • • P (a s )P -1 (c 1 ) • • • P -1 (c s ),
where the sum runs over all ways of cutting the word u = (u 1 , . . . , u r ) into 3s chunks of which the b i 's may not be empty, a 1 and c s may be empty, and the interior chunks a i and c j may be empty as long as no interior double chunk c i a i+1 is empty. Note that because GARI mu is a huge group containing all possible moulds with constant term 1, the automorphism garit(P ) cannot be determined simply by giving its value on some simple generators as we do for A p . However, garit(P ) extends to a taking the value a, and restricted to the Lie algebra (ARI a lu ) pol generated by a and B (isomorphic to Lie[a, b]), we find garit(P ) • a = a, garit(P ) • B = P BP -1 .

(A.7)

  Let C i = ad(a) i-1 (b) for i ≥ 1, and let Lie[C] denote the Lie algebra Lie[C 1 , C 2 , . . .]. It is an easy consequence of Lazard elimination that Lie[C] is a free Lie algebra on the generators C i , and that Lie[a, b] Qa ⊕ Lie[C] (see Appendix). In other words, the elements in Lie[C] are all the elements of Lie[a, b] having no linear term in a. Definition. Let Der 0 Lie[a, b] denote the subspace of derivations D ∈ Der Lie[a, b] that annihilate [a, b] and such that D(a) and D(b) lie in Lie[C]. Hain-Matsumoto and Enriquez both give derivation representations of the elliptic spaces into Der 0 Lie[a, b], but Enriquez's Lie morphism grt ell → Der 0 Lie[a, b] is injective (by [T2], cf. below for more detail), whereas Hain-Matsumoto conjecture this result in the motivic situation. However, Hain-Matsumoto compute the image of u in Der 0 Lie[a, b] and show that it is equal to a certain explicitly determined Lie algebra b 3 related to SL 2 (Z) (or to the Artin braid group B 3 on three strands), namely the Lie algebra generated by derivations 2i , i ≥ 0 defined by 2i (a) = ad(a) 2i (b),

  3. ♦Definition. A mould P is said to be ∆-bialternal if ∆ -1 (P ) is bialternal, i.e., P ∈ ∆(ARI al * al ari ). The elliptic double shuffle Lie algebra ds ell ⊂ Lie[a, b] is the set of Lie polynomials which map under ma to polynomial-valued ∆-bialternal moulds that are even in depth 1, i.e.,ds ell = ma -1 ∆ ARI ∆,al * al ari .(1.3.9) Taken together, Theorems 1.3.2 and 1.3.3 show that the image of ma(ds) = ARI pol,al * il ari under Ad ari (invpal) lies in ARI ∆,al * al ari

2. 1 .

 1 The push-invariance and extension lemmas Definition. For p ∈ Lie[a, b], write p = p a a + p b b and set

  p be two polynomials in Lie[a, b] such that the coefficient of a in p and p is zero, and let D denote the derivation of Lie[a, b] given by a → p, b → p . Then D([a, b]) = 0 if and only if p is push-invariant and p is its partner. Proof. Let P = ma(p) = ma D(a) and P = ma(p ) = ma D(b) . Using the fact that ma is a Lie algebra morphism (see Appendix) and the first identity of (1.3.2) we find that ma D([a, b] = ma [D(a), b] + [a, D(b)] = [P, B] -dur(P ), (2.1.3) where B = ma(b) is the mould concentrated in depth 1 given by B(u 1 ) = 1. Note that the mould [P, B] -dur(P ) is zero in depths r ≤ 1.

  .1.8) but this coincides with (2.1.2), showing that P is the partner of P . ♦ Lemma 2.1.2 Let D be a derivation of the Lie subalgebra Lie[t 01 , t 02 ] ⊂ Lie[a, b]. Then (i) there exists a unique derivation D ∈ Der 0 Lie[a, b] having the following two properties: (i.1) D(t 02 ) = D(t 02 ); (i.2) D(b) is the partner of D(a). (ii) If D(t 12 ) = 0 and D(a) is push-invariant, then D is the unique extension of D to all of Lie[a, b].

  E ψ is the unique extension of Ẽψ to all of Lie[a, b]. But Enriquez' derivation D ψ is an extension of Dψ to all of Lie[a, b], and it also satisfies D ψ (t 12 ) = 0, so by Lemma 2.1.1, D ψ (a) = α + = γ + (ψ) is push-invariant; thus by Lemma 2.1.2 (ii) D ψ is the unique extension of Dψ to all of Lie[a, b]. Thus, since Ẽψ = Dψ , we must have E ψ = D ψ , and in particular E ψ (a) = m = D ψ (a) = γ + (ψ). Taking ma of both sides yields the desired equality (1.3.5). ♦ 3. Proof of Proposition 2.2.1

  Lemma 3.1.1. (i) The automorphism dar extends to a taking the value dar(a) = a; (ii) The derivation dur extends to a taking the value dur(a) = 0;

  Thus the derivation E M of Lie[a, b] agrees with E ψ on a and on [a, b], so since furthermore E M (b) ∈ Lie[a, b] Lie[a], they are equal. ♦

  which proves the desired identity. ♦ Let us now define the group GARI Dgari . We start by defining the exponential map exp Dari : ARI Dari → GARI by exp Dari (P ) P ∈ ARI satisfies the equality exp Darit(P ) (a) = exp Dari (P ). (3.2.3) This map is easily seen to be invertible, since for any Q ∈ GARI we can recover P such that exp Dari (P ) = Q recursively depth by depth. Let log Dari denote the inverse of exp Dari . For each P ∈ GARI, we then define an automorphism Dgarit(P ) ∈ Aut ARI lu by Dgarit(P ) = Dgarit exp Dari log Dari (P ) = exp Darit log Dari (P ) . Finally, we define the multiplication Dgari on GARI by Dgari(P, Q) = exp Dari ch Dari (log Dari (P ), log Dari (Q)) = exp Darit(log Dari (P )) • exp Darit(log Dari (Q)) • a Dgarit(P ) • Dgarit(Q) • a = Dgarit(P ) • Q, where ch Dari denotes the Campbell-Hausdorff law on ARI Dari . We obtain the following commutative diagram, analogous to Ecalle's diagram (A.18) (cf. Appendix): ARI Dari exp Dari / / Darit GARI Dgari Dgarit Der ARI lu exp / / Aut ARI lu .

  in diagram (3.2.6) is explicitly given by the formula ∆ * (Q) = 1 -dar du inv gari (Q) .(3.2.9)Proof. Let Q ∈ GARI, and set P = log ari (Q). Let R = exp ari (-P ). By Lemma A.1 from the Appendix, the derivation -arit(P ) + ad(P ) extends to a taking the value [a, P ] on a, and we have exp -arit(P ) + ad(P ) • a = R -1 a R. (3.2.10) By (3.1.1), we have exp Darit ∆(P ) = dar • exp -arit(P ) + ad(P ) • dar -1 .Recall that dar(a) = a by Lemma 3.1.1 (i), and dar is an automorphism of ARI a lu ; in particular du commutes with dar. Thus we haveexp Darit ∆(P ) • a = dar • exp -arit(P ) + ad(P ) • a = dar(R -1 a R) by Lemma A.1 = dar(R) -1 a dar(R) = a -du dar(R) by (3.2.8) = a -dar duR . (3.2.11) Now, using P = log ari (Q), we compute ∆ * (Q) = 1 -a + Dgarit ∆ * (Q) • a by (3.2.5) = 1 -a + Dgarit exp Dari ∆(log ari (Q)) • a by (3.2.6) = 1 -a + Dgarit exp Dari ∆(P ) • a = 1 -a + expDarit ∆(P ) • a by (3.2.4) = 1 -dar du exp ari (-P ) by (3.2.11) = 1 -dar du inv gari (Q) .

  We have the identity ∆ * (invpal) = ma 1 -a + Ber -b (a) .(3.2.13) Proof. Applying (3.2.9) to Q = invpal = inv gari (pal), we find∆ * (invpal) = 1 -dar dupal , (3.2.14)where dupal is the mu-dilator of pal given in (1.3.3), discovered by Ecalle. Comparing the elementary mould identity ma ad(-b) r (-a) = shows that dar(dupal) is given in depth r ≥ 1 by dar(dupal)(u 1 , . . . , u r ) -b) r (-a) . Since the constant term of dar dupal (∅) is 0, this yields dar dupal = ma Ber -b (-a) + a = ma a -Ber -b (a) , so (3.2.14) implies the desired identity (3.2.13). ♦ 3.3. Proof of Proposition 2.2.1 Let ψ ∈ grt. We return to the notation of (1.3.8). By Corollary 3.1.4, we have a derivation E M = E ψ ∈ Der Lie[a, b] obtained by restricting the derivation E ψ = Darit(M ) to the Lie subalgebra of ARI a lu generated by a and B = ma(b), which is precisely (ARI a lu ) pol,al , and transporting the derivation to the isomorphic space Lie[a, b]. The purpose of this section is to prove (2.2.1), i.e., E ψ (t 02 ) = [ψ(t 02 , t 12 ), t 02 ].

  = P Q -Q P , gives ARI the structure of a Lie algebra that we call ARI lu . Mould symmetries. A mould P is said to be alternal if u∈sh (u 1 ,...,u i ),(u i+1 ,...,u r ) i ≤ r -1. It is well-known that p ∈ Q C satisfies the shuffle relations if and only if p is a Lie polynomial, i.e., p ∈ Lie[C]. The alternality property on moulds is analogous to these shuffle relations, i.e., a polynomial p ∈ Q C satisfies the shuffle relations if and only if ma(p) is alternal. (See[START_REF] Schneps | An Introduction to Ecalle's theory of moulds[END_REF] §2.3 and Lemma 3.4.1.].) This shows that, writing ARI al for the subspace of alternal moulds and ARI pol,al for the subspace of alternal polynomial-valued moulds, the map ma restricts to a Lie algebra isomorphism ma : Lie[C] ma -→ ARI pol,al lu .

  The relation between the two is based on the fact that Lie[x, y] is identified with the Lie algebra of the fundamental group of the thrice-punctured sphere, whereas Lie[a, b] is identified with the Lie algebra of the once-punctured torus. The natural Lie morphism Lie[x, y] → Lie[a, b], reflecting the underlying topology, is given by x → t 01 , y → t 02 , where we write Ber x = ad(x)/ exp(ad(x)) -1 for any x ∈ Lie[a, b], and set t 01 = Ber b (-a), t 02 = Ber -b (a). We show that certain derivations of Lie[x, y], transported to the free Lie subalgebra Lie[t 01 , t 02 ] ⊂ Lie[a, b] have a unique extension to derivations of all of Lie

  Theorem 1.3.2. (cf. [E] 5 ) The adjoint map Ad ari (invpal) induces a Lie isomorphism of Lie subalgebras of ARI ari : Ad ari (invpal) : ARI

	al * il ari	∼ → ARI ari , al * al	(1.3.6)

and if F ∈ ARI al * il and C is the constant mould such that swap(F + C) is alternil, then swap Ad ari (invpal)(F ) + C is alternal, i.e., the constant corrections for F and Ad ari (invpal) • F are the same. In particular if C = 0, i.e., if F is al/il, then Ad ari (invpal)(F ) lies in al/al.

  • B 1 = 0 for all Q ∈ ARI shown in Lemma 3.1.1 (iv). ♦Finally, we set ∆ * = exp Dari • ∆ • log ari , to obtain the commutative diagram of isomorphisms a special role in the proof of Proposition 2.2.1. Indeed, the key result in our proof Proposition 2.2.1 is an explicit formula for the map ∆ * . In order to formulate it, we first define the mu-dilator of a mould, introduced by Ecalle in[E2].Definition. Let P ∈ GARI. Then the mu-dilator of P , denoted duP , is defined by Ecalle writes this in the equivalent form dur(P ) = P duP , and by (3.1.3), this means that [P, a] = P a -aP = P duP = P , whch multiplying by P -1 , gives us the useful formulation 6

	ARI ari	∆ -→ ARI Dari
	exp ari ↓		↓ exp Dari
	GARI gari	∆ * → GARI Dgari ,	(3.2.6)
	which will play duP = P -1 dur(P ).	(3.2.7)
	P -1 aP = a -duP.	(3.2.8)
	Proposition 3.2.3. The isomorphism
	∆

* : GARI gari → GARI Dgari

2i([a, b]) = 0 1 , whereas Enriquez considers the same Lie algebra b 3 , shows that it injects into r ell , and conjectures that they are equal 2 .1 This Lie algebra was introduced by Tsunogai in [T1, §3] (see also[P],[BS] and[START_REF] Brown | Depth-graded motivic multiple zeta values[END_REF] for some results on its interesting structure. The 2i also play an important role in[CEE] and[START_REF] Enriquez | Elliptic Associators[END_REF].2 It is really remarkable that these two papers were written totally independently of one another.

With respect to the notation of[START_REF] Enriquez | Elliptic Associators[END_REF] we have Lie[a, b] = t 1,2 , a = y 1 = x - 1 , b = x 1 = x + 1 (Enriquez uses both notations).

This result is stated and used constantly in[E], as well as many other analogous results concerning other symmetries. But the proof is not given. Ecalle was kind enough to send us a sketch of the proof in a personal letter, relying on the fundamental identity (2.62) of [E], itself not proven there. Full details of the reconstructed proof can be found in[S], with (2.62) proved in Theorem 2.8.1 and Theorem 1.3.2 above proved in §4.6.

We are grateful to B. Enriquez for spotting this enlightening interpretation of the mu-dilator, which cannot even be stated meaningfully for general moulds unless a is added to ARI.

Ecalle uses the notation ARI for the space of these moulds equipped with the ari-bracket, that we denote ARI ari , and in fact he considers more general bimoulds in two sets of variables.

Proof. (i) Let T = D(t 02 ), and write T = n≥w T n for its homogeneous parts of weight n, where the weight is the degree as a polynomial in a and b, and w is the minimal weight occurring in T . We will construct a derivation D satisfying D(t 02 ) = D(t 02 ) via the equality [b, [b, [b, a]]]] [b, [b, [b, a]]] -1 720 [b, [D(b), [b, [b, a]]]] -1 720 [b, [b, [D(b), [b, a]]]] + • • • .

(2.1.9)

We construct D(a) by solving (2.1.9) in successive weights starting with w. We start by setting D(a) w = T w and D(a) w+1 = T w+1 , and take D(b) w and D(b) w+1 to be their partners. We then continue to solve the successive weight parts of (2.1.9) for D(a) in terms of T and lower weight parts of D(b). For instance the next few steps after weights w and w + 1 are given by

In For each ψ ∈ grt, let f (x, y) = ψ(x, -y). Let A = Ad ari (invpal) • ma(f ) as before, and M = ∆(A). By Corollary 1.3.4, there exists a polynomial m ∈ Lie[C] such that

Since by the same corollary m is push-invariant, we see that by Lemma 2.1.1 there exists a unique derivation

The derivation E P has the property that the values E P (a) and

Proof. By the isomorphism (3.1.4), every mould P ∈ (ARI a lu ) pol,al has a unique preimage in Lie [a, b] under ma: we write p = ma -1 (P ). Recall that B = ma(b). By assumption, P is an alternal polynomial-valued mould, and so is Darit(P ) • B since P preserves such moulds. Thus we can define E P by setting E P (a) = ma -1 (P ), E P (b) = ma -1 Darit(P ) • B . In particular this means that the monomial a does not appear in the polynomials E P (a) and E P (b). ♦ Lemma 3.1.3. Let P be an alternal polynomial-valued mould. Then Darit(P ) preserves (ARI a lu ) pol,al if and only if P is push-invariant.

Proof. By the isomorphism (3.1.4), (ARI a lu ) pol,al is generated as a Lie algebra under the lu bracket by ma(a) = a and ma(b) = B. Since Darit(P ) • a = P is alternal and polynomial-valued by assumption, it suffices to determine when Darit(P )•B is alternal and polynomial. Let N = ∆ -1 P , and set B -1 = dar -1 (B), so B -1 is concentrated in depth 1 with B -1 (u 1 ) = 1/u 1 . We compute

In order for this mould to be polynomial-valued, it is necessary and sufficient that the numerator should be zero when

But the right-hand term is equal to push -1 (P ), so this condition is equivalent to the push-invariance of P . ♦ Corollary 3.1.4. The derivation E ψ defined in section 2.2 is equal to the derivation E M associated to Darit(M ) as in Lemma 3.1.2.

Appendix: Mould basics

For the purposes of this article, we use the term "mould" to refer only to rational-function valued moulds with coefficients in Q; thus, a mould is a family of functions {P r (u 1 , . . . , u r ) | r ≥ 0} with P r (u 1 , . . . , u r ) ∈ Q(u 1 , . . . , u r ). In particular P 0 (∅) is a constant. The depth r part of a mould is the function P r (u 1 , . . . , u r ) in r variables. By defining addition and scalar multiplication of moulds in the obvious way, i.e., depth by depth, moulds form a Q-vector space that we call M oulds. Following Ecalle, we often drop the subscript r from the mould notation; i.e., we write P (u 1 , . . . , u r ) to mean the rational function P r (u 1 , . . . , u r ), where the number of variables automatically indicates which depth part we are considering.

We write GARI for the set of moulds with P (∅) = 1, and ARI for the set of moulds 8 with P (∅) = 0. Then ARI forms a vector subspace of M oulds.

Let (M oulds) pol denote the subspace of polynomial-valued moulds, i.e., moulds such that P (u 1 , . . . , u r ) is a polynomial in each depth r ≥ 0, and ARI pol the polynomial-valued subspace of ARI. In this appendix we will stress the connections between polynomial-valued moulds and power series in the non-commutative variables a and b, showing in particular how familiar notions from multizeta theory (the Poisson-Ihara bracket, the twisted Magnus group etc.) not only translate over to the corresponding moulds, but generalize to all moulds.

Let C i = ad(a) i-1 (b) for i ≥ 1. Let the depth of a monomial C i 1 • • • C i r be the number r of C i in the monomial; the depth forms a grading on the free associative ring of polynomials in the

. . denote the depth completion of this ring, i.e., Q C is the space of power series that are polynomials in each depth. We also write

for the corresponding free Lie algebra. Note that the freeness follows from Lazard elimination, which also shows that we have the isomorphism

Ecalle uses the notation ma to denote the standard vector space isomorphism from Q C to the space (M oulds) pol of polynomial-valued moulds defined by

on monomials and extended by linearity. This map ma can also be considered as a ring isomorphism when (M oulds) pol is equipped with the suitable multiplication, cf. the remarks following (A.4) below. (We use the same notation ma when C i = ad(x) i-1 (y), for polynomials usually considered in Lie[x, y], such as polynomials in grt.) For any map Φ : Q C → Q C , we define its transport ma(Φ) to (M oulds) pol , namely the corresponding map on polynomial-valued moulds ma(Φ) : (M oulds) pol → (M oulds) pol by the obvious relation

In analogy with the formula for given in (A.6), garit defines a multiplication law gari on GARI by the formula

We write GARI gari for the group obtained by equipping GARI with this multiplication. Poisson-Ihara bracket, exponential, linearization. For all P ∈ ARI, Ecalle defines a derivation arit(P ) of ARI lu by the formula , and as we saw in Lemma 3.1.1 (iii), it extends to all of (ARI a lu ) pol,al taking the value 0 on a. It corresponds on the isomorphic Lie algebra Lie [a, b] to the Ihara derivation D f defined by

(A.9)

The Lie bracket {•, •} that we put on L[C], known as the Poisson bracket or Ihara bracket, comes from bracketing the derivations D f , i.e.,

We obtain a pre-Lie law by linearizing the multiplication law defined in (A.6). In fact, because is linear in p, we only need to linearize q, so we write q = 1 + tf and compute the coefficient of t in All these standard constructions extend to the case of general moulds; Ecalle gives explicit formulas for the pre-Lie law preari and for the exponential exp ari , namely preari(P, Q) = P Q + arit(Q) We conclude this appendix with a linearization lemma used in the proof of Proposition 3.2.3.