The map from

a Lie algebra under the Poisson bracket.

Proof. By the definition of𝑎𝑟𝑖,𝐴𝑅𝐼^𝑝𝑜𝑙is a Lie subalgebra of𝐴𝑅𝐼. Also, Lemma III.3.10 shows that the space𝐴𝑅𝐼_{𝑐𝑖𝑟𝑐𝑛𝑒𝑢𝑡}of circ-neutral moulds is a Lie subalgebra of𝐴𝑅𝐼_{∗𝑐𝑖𝑟𝑐𝑛𝑒𝑢𝑡}.

Thus𝐴𝑅𝐼^Δ

𝑎𝑙+𝑝𝑢𝑠ℎ∕𝑐𝑖𝑟𝑐𝑛𝑒𝑢𝑡is a Lie algebra inside𝐴𝑅𝐼𝑎𝑙+𝑝𝑢𝑠ℎ∗𝑐𝑖𝑟𝑐𝑛𝑒𝑢𝑡^Δ . So the intersection 𝐴𝑅𝐼^𝑝𝑜𝑙∩𝐴𝑅𝐼𝑎𝑙+𝑝𝑢𝑠ℎ∕𝑐𝑖𝑟𝑐𝑛𝑒𝑢𝑡^Δ =𝐴𝑅𝐼𝑎𝑙+𝑝𝑢𝑠ℎ∕𝑐𝑖𝑟𝑐𝑛𝑒𝑢𝑡^𝑝𝑜𝑙

is one as well.

III.3.3 The map from𝔨𝔯𝔳→𝔨𝔯𝔳_𝑒𝑙𝑙

In this subsection we prove our next main result on the elliptic Kashiwara-Vergne Lie algebra, which is analogous to known results on the elliptic Grothendieck-Teichmüller Lie algebra of [E] and the elliptic double shuffle Lie algebra of [S3]. The subsection III.3.4 below is devoted to connections between these three situations.

Theorem III.3.12. ([RS1])There is an injective Lie algebra morphism

𝔨𝔯𝔳↪𝔨𝔯𝔳_𝑒𝑙𝑙 (III.3.13)

The proof constructs the morphism from𝔨𝔯𝔳to𝔨𝔯𝔳_𝑒𝑙𝑙in four main steps as follows.

Step 1. We first consider a twisted version of the Kashiwara-Vergne Lie algebra, or rather of the associated polynomial space𝑉_𝔨𝔯𝔳of Definition II.3.11, via the map

𝜈 ∶𝑉_𝔨𝔯𝔳→^∼ 𝑊_𝔨𝔯𝔳 (III.3.14)

𝑓 ↦𝜈(𝑓), (III.3.15)

where𝜈is the automorphism ofℚ⟨𝑥, 𝑦⟩defined by

𝜈(𝑥) =𝑧= −𝑥−𝑦, 𝜈(𝑦) =𝑦. (III.3.16) In paragraph III.3.3, we prove that𝑊_𝔨𝔯𝔳is a Lie algebra under the Poisson or Ihara bracket, and give a description of𝑊_𝔨𝔯𝔳via two properties, the “twisted” versions of the two defining properties of𝑉_𝔨𝔯𝔳 given in Definition II.3.11.

Step 2.In paragraph III.3.3, we study the mould space𝑚𝑎( 𝑊_𝔨𝔯𝔳)

. Thanks to the compatibility of the 𝑎𝑟𝑖-bracket with the Poisson bracket, this space is a Lie subalgebra of𝐴𝑅𝐼_𝑎𝑟𝑖. Just as we reformulated the defining properties of𝔩𝔨𝔳in mould terms, proving that𝑚𝑎(𝔩𝔨𝔳) = 𝐴𝑅𝐼𝑎𝑙+𝑝𝑢𝑠ℎ∕𝑐𝑖𝑟𝑐𝑛𝑒𝑢𝑡^𝑝𝑜𝑙 , here we reformulate the defining properties of𝑊_𝔨𝔯𝔳in mould terms: explicitly, we show that

𝑚𝑎( 𝑊_𝔨𝔯𝔳)

=𝐴𝑅𝐼𝑎𝑙+𝑠𝑒𝑛∗𝑐𝑖𝑟𝑐𝑐𝑜𝑛𝑠𝑡^𝑝𝑜𝑙 , (III.3.17)

the space of polynomial-valued moulds that are alternal, satisfy a certainsenary relation (III.3.23) introduced by Écalle (see below), and whose swap is circ-constant up to addition of a constant-valued mould.

We observe that if 𝐵 ∈ 𝐴𝑅𝐼 is a polynomial-valued mould of homogeneous degree𝑛 whose swap is circ-constant up to addition of a constant-valued mould, then the constant-valued mould𝐵₀is uniquely determined as being the mould concentrated in depth𝑛and taking the value𝑐∕𝑛there, where 𝐵(𝑣₁) =𝑐𝑣^𝑛−1

1 .

Step 3.For this part, we use again the mould𝑝𝑎𝑙previously defined in (II.2.2) and the adjoint operator 𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙)on𝐴𝑅𝐼_𝑎𝑟𝑖.

LettingΞdenote the map

𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙)◦𝑝𝑎𝑟𝑖∶𝐴𝑅𝐼_𝑎𝑟𝑖→𝐴𝑅𝐼_𝑎𝑟𝑖, we show that it yields an injective Lie morphism

Ξ ∶𝐴𝑅𝐼𝑎𝑙+𝑠𝑒𝑛∗𝑐𝑖𝑟𝑐𝑐𝑜𝑛𝑠𝑡^𝑝𝑜𝑙 →𝐴𝑅𝐼𝑎𝑙+𝑝𝑢𝑠ℎ∗𝑐𝑖𝑟𝑐𝑛𝑒𝑢𝑡^Δ (III.3.18) of subalgebras of𝐴𝑅𝐼_𝑎𝑟𝑖.

Step 4. The final step is to compose (III.3.18) with the Lie morphism Δ ∶ 𝐴𝑅𝐼_𝑎𝑟𝑖 → 𝐴𝑅𝐼_{𝐷𝑎𝑟𝑖}, obtaining an injective Lie morphism

𝐴𝑅𝐼𝑎𝑙+𝑠𝑒𝑛∗𝑐𝑖𝑟𝑐𝑐𝑜𝑛𝑠𝑡^𝑝𝑜𝑙 →Δ(

𝐴𝑅𝐼𝑎𝑙+𝑝𝑢𝑠ℎ∗𝑐𝑖𝑟𝑐𝑛𝑒𝑢𝑡^Δ

),

where the left-hand space is a subalgebra of𝐴𝑅𝐼_𝑎𝑟𝑖and the right-hand one of 𝐴𝑅𝐼_{𝐷𝑎𝑟𝑖}. Since the right-hand space is equal to𝑚𝑎(𝔨𝔯𝔳_𝑒𝑙𝑙), the desired injective Lie morphism𝔨𝔯𝔳→ 𝔨𝔯𝔳_𝑒𝑙𝑙is obtained by composing all the maps described above, as shown in the following diagram:

𝔨𝔯𝔳 by(II.3.2)↓

𝑉_𝔨𝔯𝔳 by(III.3.14)↓ 𝜈

𝑊_𝔨𝔯𝔳 𝔨𝔯𝔳_𝑒𝑙𝑙

by(III.3.17)↓ 𝑚𝑎 𝑚𝑎⁻¹ ↑by(III.3.8)

𝐴𝑅𝐼𝑎𝑙+𝑠𝑒𝑛∗𝑐𝑖𝑟𝑐𝑐𝑜𝑛𝑠𝑡^𝑝𝑜𝑙 Ξ

⟶𝐴𝑅𝐼𝑎𝑙+𝑝𝑢𝑠ℎ∗𝑐𝑖𝑟𝑐𝑛𝑒𝑢𝑡^Δ Δ

⟶Δ(

𝐴𝑅𝐼𝑎𝑙+𝑝𝑢𝑠ℎ∗𝑐𝑖𝑟𝑐𝑛𝑒𝑢𝑡^Δ

) by(III.3.18)

Step 1: The twisted space𝑊_𝔨𝔯𝔳

Proposition III.3.13. Let𝑊_𝔨𝔯𝔳=𝜈(𝑉_𝔨𝔯𝔳). Then𝑊_𝔨𝔯𝔳is a Lie algebra under the Poisson bracket.

Proof. The key point is the following lemma on derivations.

Lemma III.3.14. Conjugation by𝜈induces an isomorphism of Lie algebras

𝔰𝔡𝔢𝔯₂→^∼ 𝔦𝔡𝔢𝔯₂ (III.3.19)

𝐸_𝑎,𝑏↦𝑑_𝜈(𝑏).

Proof. Recall that 𝐸_𝑎,𝑏 ∈ 𝔰𝔡𝔢𝔯₂ maps𝑥 ↦ [𝑥, 𝑎] and𝑦 ↦ [𝑦, 𝑏], and𝑑_𝜈(𝑏) ∈ 𝔦𝔡𝔢𝔯₂ is the Ihara derivation defined by𝑥↦0,𝑦↦[𝑦, 𝜈(𝑏)].

Let us first show that 𝑑_𝜈(𝑏) is the conjugate of𝐸_𝑎,𝑏 by 𝜈, i.e. 𝑑_𝜈(𝑏) = 𝜈◦𝐸_𝑎,𝑏◦𝜈 (since 𝜈 is an involution). It is enough to show they agree on𝑥and𝑦, so we compute

𝜈◦𝐸_𝑎,𝑏◦𝜈(𝑥) =𝜈◦𝐸_𝑎,𝑏(𝑧) = 0 =𝑑_𝜈(𝑏)(𝑥) and

𝜈◦𝐸_𝑎,𝑏◦𝜈(𝑦) =𝜈◦𝐸_𝑎,𝑏(𝑦) =𝜈( [𝑦, 𝑏])

= [𝑦, 𝜈(𝑏)] =𝑑_𝜈(𝑏)(𝑦).

This shows that 𝜈◦𝐸_𝑎,𝑏◦𝜈 is indeed equal to𝑑_𝜈(𝑏). To show that 𝑑_𝜈(𝑏) lies in 𝔦𝔡𝔢𝔯₂, we check that 𝑑_𝜈(𝑏)(𝑧)is a bracket of𝑧with another element of𝔩𝔦𝔢₂:

𝑑_𝜈(𝑏)(𝑧) =𝜈◦𝐸_𝑎,𝑏◦𝜈(𝑧) =𝜈◦𝐸_𝑎,𝑏(𝑥) =𝜈([𝑥, 𝑎]) = [𝑧, 𝜈(𝑎)].

The same argument goes the other way to show that conjugation by𝜈 maps an element of𝔦𝔡𝔢𝔯₂to an element of𝔰𝔡𝔢𝔯₂, which yields the isomorphism (III.3.19) as vector spaces. To see that it is also an isomorphism of Lie algebras, it suffices to note that conjugation by𝜈 preserves the Lie bracket of derivations in𝔡𝔢𝔯₂, i.e.

𝜈◦[𝐷₁, 𝐷₂]◦𝜈 = [𝜈◦𝐷₁◦𝜈, 𝜈◦𝐷₂◦𝜈],

since𝜈 is an involution. Since the Lie brackets on𝔰𝔡𝔢𝔯₂and𝔦𝔡𝔢𝔯₂are just restrictions to those sub-spaces of the Lie bracket on the space of all derivations, conjugation by𝜈carries one to the other.

We use the lemma to complete the proof of Proposition III.3.13. Write 𝔨𝔯𝔳^𝜈 = {𝜈◦𝐸◦𝜈|𝐸∈𝔨𝔯𝔳}⊂𝔦𝔡𝔢𝔯₂.

By restricting the isomorphism (III.3.19) to the subspace 𝔨𝔯𝔳 ⊂ 𝔰𝔡𝔢𝔯₂, we obtain a commutative diagram of isomorphisms of vector spaces

𝔨𝔯𝔳 → 𝔨𝔯𝔳^𝜈,

↓ ↓ 𝑉_𝔨𝔯𝔳→^𝜈 𝑊_𝔨𝔯𝔳,

where the left-hand vertical arrow is the isomorphism (II.3.2) mapping𝐸_𝑎,𝑏↦𝑏, and the right-hand vertical map sends an Ihara derivation𝑑_𝑓 to𝑓.

Equipping𝑊_𝔨𝔯𝔳with the Lie bracket inherited from𝔨𝔯𝔳^𝜈 makes this into a commutative diagram of Lie isomorphisms. But this bracket is nothing other than the Poisson bracket since𝔨𝔯𝔳^𝜈 ⊂𝔦𝔡𝔢𝔯₂.

We now give a characterization of 𝑊_𝔨𝔯𝔳 by two defining properties which are the twists by𝜈 of those defining𝑉_𝔨𝔯𝔳. Recall that𝛽 is the the backwards operator.

Proposition III.3.15. The space𝑊_𝔨𝔯𝔳is the space spanned by polynomials𝑏∈𝔩𝔦𝔢_𝐶, of homogeneous degree𝑛≥3, such that

(i)𝑏_𝑦−𝑏_𝑥isanti-palindromic, i.e.𝛽(𝑏_𝑦−𝑏_𝑥) = (−1)^𝑛−1(𝑏_𝑦−𝑏_𝑥), and (ii)𝑏+ ^𝑐

𝑛𝑦^𝑛iscirc-constant, where𝑐= (𝑏|𝑥^𝑛−1𝑦).

Proof. Let𝑓 = 𝜈(𝑏), so that𝑓 ∈ 𝑉_𝔨𝔯𝔳. Then the property that 𝑏_𝑦 −𝑏_𝑥is anti-palindromic is precisely equivalent to the push-invariance of𝑓 (this is proved as the equivalence of properties (iv) and (v) of Theorem 2.1 of [S1]). This proves (i).

For (ii), we note that since𝑓 ∈ 𝑉_𝔨𝔯𝔳, 𝑓^𝑦−𝑓^𝑥is push-constant for the value𝑐 = (𝑓|𝑥^𝑛−1𝑦) = (−1)^𝑛−1(𝑏|𝑥^𝑛−1𝑦). We have

𝑏(𝑥, 𝑦) =𝑥𝑏^𝑥(𝑥, 𝑦) +𝑦𝑏^𝑦(𝑥, 𝑦), so

𝑓(𝑥, 𝑦) =𝑏(𝑧, 𝑦) =𝑧𝑏^𝑥(𝑧, 𝑦) +𝑦𝑏^𝑦(𝑧, 𝑦) = −𝑥𝑏^𝑥(𝑧, 𝑦) −𝑦𝑏^𝑥(𝑧, 𝑦) +𝑦𝑏^𝑦(𝑧, 𝑦).

Thus since𝑓(𝑥, 𝑦) =𝑥𝑓^𝑥(𝑥, 𝑦) +𝑦𝑓^𝑦(𝑥, 𝑦), this gives

𝑓^𝑥= −𝑏^𝑥(𝑧, 𝑦) and 𝑓^𝑦 = −𝑏^𝑥(𝑧, 𝑦) +𝑏^𝑦(𝑧, 𝑦), so

𝑓^𝑦−𝑓^𝑥=𝑏^𝑦(𝑧, 𝑦) =𝜈(𝑏^𝑦).

Thus to prove the result, it suffices to prove that the following statement: if𝑔∈ℚ⟨𝐶⟩is a polynomial of homogeneous degree𝑛that is push-constant for(−1)^𝑛−1𝑐, then𝜈(𝑔) is push-constant for𝑐, since taking𝑔 = 𝑓^𝑦−𝑓^𝑥then shows that𝜈(𝑔) =𝑏^𝑦is push-constant for𝑐. The proof of this statement is straightforward using the substitution𝑧= −𝑥−𝑦(but see the proof of Lemma 3.5 in [S1] for details).

Since𝑐 = 0if𝑓 ∈𝑉_𝔨𝔯𝔳is of even degree𝑛, this proves (ii).

Step 2: The mould version𝑚𝑎(𝑊_𝔨𝔯𝔳)

The space𝑚𝑎(𝑊_𝔨𝔯𝔳)is closed under the 𝑎𝑟𝑖-bracket by (??), since𝑊_𝔨𝔯𝔳is closed under the Poisson bracket.

Let𝑏∈ 𝑊_𝔨𝔯𝔳and let𝐵 = 𝑚𝑎(𝑏). Then since𝑏is a Lie polynomial,𝐵 is an alternal polynomial mould. Let us give the mould reformulations of properties (i) and (ii) of Proposition III.3.15. The second property is easy since we already showed, in Proposition II.3.22, that a polynomial𝑏is circ-constant if and only if𝑠𝑤𝑎𝑝(𝐵)is circ-constant.

Expressing the first property in terms of moulds is more complicated and calls for an identity discovered by Écalle. We need to use the mould operators𝑚𝑎𝑛𝑡𝑎𝑟and𝑝𝑎𝑟𝑖defined by

𝑚𝑎𝑛𝑡𝑎𝑟(𝐵) = (𝑢₁,…, 𝑢_𝑟) = (−1)^𝑟−1𝐵(𝑢_𝑟, 𝑢_𝑟−1,…, 𝑢₁) (III.3.20) 𝑝𝑎𝑟𝑖(𝐵)(𝑢₁,…, 𝑢_𝑟) = (−1)^𝑟𝐵(𝑢₁,…, 𝑢_𝑟). (III.3.21) The operator𝑝𝑎𝑟𝑖extends the operator𝑦↦−𝑦on polynomials to all moulds, and𝑚𝑎𝑛𝑡𝑎𝑟extends the operator𝑓 ↦ (−1)^𝑛−1𝛽(𝑓). Above all, we need Écalle’s mould operator𝑡𝑒𝑟𝑢, defined by taking the mould𝑡𝑒𝑟𝑢(𝐵)to be equal to𝐵 in depths0and1, and for depths𝑟 >1, setting

𝑡𝑒𝑟𝑢(𝐵)(𝑢₁,…, 𝑢_𝑟) = (III.3.22)

𝐵(𝑢₁,…, 𝑢_𝑟) + 1 𝑢_𝑟

(

𝐵(𝑢₁,…, 𝑢_𝑟−2, 𝑢_𝑟−1+𝑢_𝑟) −𝐵(𝑢₁,…, 𝑢_𝑟−2, 𝑢_𝑟−1)) . Lemma III.3.16. Let𝑏∈𝔩𝔦𝔢_𝐶. Then the following are equivalent:

(1)𝑏_𝑦−𝑏_𝑥is anti-palindromic;

(2)𝐵=𝑚𝑎(𝑏)satisfies thesenary relation

𝑡𝑒𝑟𝑢◦𝑝𝑎𝑟𝑖(𝐵) =𝑝𝑢𝑠ℎ◦𝑚𝑎𝑛𝑡𝑎𝑟◦𝑡𝑒𝑟𝑢◦𝑝𝑎𝑟𝑖(𝐵). (III.3.23) Proof. The statement is a consequence of the following result, proved in A.3 of the Appendix of [S1]. Let ̃𝑏 ∈ 𝔩𝔦𝔢_𝐶 and let 𝐵̃ = 𝑚𝑎(̃𝑏). Write ̃𝑏 = ̃𝑏_𝑥𝑥+ ̃𝑏_𝑦𝑦as usual. Then for each depth part (̃𝑏_𝑥+̃𝑏_𝑦)^𝑟of the polynomial ̃𝑏_𝑥+̃𝑏_𝑦(1≤𝑟≤𝑛− 1), the anti-palindromic property

(𝑓̃_𝑥+𝑓̃_𝑦)^𝑟= (−1)^𝑛−1𝛽(𝑓̃_𝑥+𝑓̃_𝑦)^𝑟 (III.3.24) translates directly to the following relation on𝐵̃:

𝑡𝑒𝑟𝑢(𝐵)(𝑢̃ ₁,…, 𝑢_𝑟) =𝑝𝑢𝑠ℎ◦𝑚𝑎𝑛𝑡𝑎𝑟◦𝑡𝑒𝑟𝑢(𝐵)(𝑢̃ ₁,…, 𝑢_𝑟). (III.3.25) Let us deduce the equivalence of (1) and (2) from that of (III.3.24) and (III.3.25). Let̃𝑏be defined by

̃𝑏(𝑥, 𝑦) =𝑏(𝑥,−𝑦). This implies that𝑏_𝑥= (−1)^𝑟̃𝑏_𝑥,𝑏_𝑦 = (−1)^𝑟−1̃𝑏_𝑦, and𝐵̃ =𝑝𝑎𝑟𝑖(𝐵). Thus𝑏_𝑦−𝑏_𝑥is anti-palindromic if and only if̃𝑏_𝑦+ ̃𝑏_𝑥is, i.e. if and only if (III.3.24) holds for ̃𝑏, which is the case if and only if (III.3.25) holds for𝐵̃, which is equivalent to (III.3.23) with𝐵̃ =𝑝𝑎𝑟𝑖(𝐵). This proves the lemma.

The following proposition summarizes the mould reformulations of the defining properties (i) and (ii) of𝑊_𝔨𝔯𝔳.

Proposition III.3.17. Let𝐴𝑅𝐼𝑎𝑙+𝑠𝑒𝑛∗𝑐𝑖𝑟𝑐𝑐𝑜𝑛𝑠𝑡^𝑝𝑜𝑙 denote the space of alternal polynomial-valued moulds satisfying the senary relation (III.3.23) and having swap that is circ-constant up to addition of a constant-valued mould. Then we have the isomorphism of Lie algebras

𝑚𝑎∶𝑊_𝔨𝔯𝔳⟶^∼ 𝐴𝑅𝐼𝑎𝑙+𝑠𝑒𝑛∗𝑐𝑖𝑟𝑐𝑐𝑜𝑛𝑠𝑡^𝑝𝑜𝑙 ⊂ 𝐴𝑅𝐼_𝑎𝑟𝑖.

Mould background: the𝑔𝑎𝑛𝑖𝑡automorphism and Ecalle’s fundamental identity The next stage of our proof, the construction of a Lie algebra morphism

𝐴𝑅𝐼^𝑝𝑜𝑙

𝑎𝑙+𝑠𝑒𝑛∗𝑐𝑖𝑟𝑐𝑐𝑜𝑛𝑠𝑡→𝐴𝑅𝐼𝑎𝑙+𝑝𝑢𝑠ℎ∗𝑐𝑖𝑟𝑐𝑛𝑒𝑢𝑡^Δ , (III.3.26) is the most difficult, and requires some further definitions .

Definition III.3.18. For any mould𝑄 ∈ 𝐺𝐴𝑅𝐼, we define an automorphism 𝑔𝑎𝑛𝑖𝑡(𝑄) of the Lie algebra𝐴𝑅𝐼_𝑙𝑢Set𝐯= (𝑣₁,…, 𝑣_𝑟), and let𝑊_𝐯denote the set of decompositions𝑑_𝐯of𝐯into chunks

𝑑_𝐯=𝐚₁𝐛₁⋯𝐚_𝑠𝐛_𝑠 (III.3.27)

for𝑠 ≥ 1, where with the possible exception of𝐛_𝑠, the 𝐚_𝑖and𝐛_𝑖are non-empty. Thus for instance, when 𝑟 = 2 there are two decompositions in 𝑊_𝐯, namely 𝐚₁ = (𝑣₁, 𝑣₂) and𝐚₁𝐛₁ = (𝑣₁)(𝑣₂), and when 𝑟 = 3 there are four decompositions, three for𝑠 = 1: 𝐚₁ = (𝑣₁, 𝑣₂, 𝑣₃), 𝐚₁𝐛₁ = (𝑣₁, 𝑣₂)(𝑣₃), 𝐚₁𝐛₁= (𝑣₁)(𝑣₂, 𝑣₃), and one for𝑠= 2:𝐚₁𝐛₁𝐚₂= (𝑣₁)(𝑣₂)(𝑣₃).

Écalle’s explicit expression for𝑔𝑎𝑛𝑖𝑡(𝑄)is given by (𝑔𝑎𝑛𝑖𝑡(𝑄)⋅𝑇)

(𝐯) = ∑

𝐚₁𝐛₁⋯𝐚_𝑠𝐛_𝑠∈𝑊_𝐯

𝑄(⌊𝐛₁)⋯𝑄(⌊𝐛_𝑠)𝑇(𝐚₁⋯𝐚_𝑠), (III.3.28) where if𝐛_𝑖is the chunk(𝑣_𝑘, 𝑣_𝑘+1,…, 𝑣_𝑘+𝑙), then we use the notation

⌊𝐛_𝑖= (𝑣_𝑘−𝑣_𝑘−1, 𝑣_𝑘+1−𝑣_𝑘−1,…, 𝑣_𝑘+𝑙−𝑣_𝑘−1). (III.3.29) We are now ready to introduce the fundamental identity of Écalle, which is the key to the con-struction of the desired map (III.3.26).

Definition III.3.19. Let constants𝑐_𝑟∈ℚ,𝑟≥1, be defined by setting𝑓(𝑥) = 1 −𝑒^−𝑥and expanding 𝑓_∗(𝑥) =∑

𝑟≥1𝑐_𝑟𝑥^𝑟+1, where𝑓_∗(𝑥)is theinfinitesimal generatorof𝑓(𝑥), defined by 𝑓(𝑥) =(

𝑒𝑥𝑝(

𝑓_∗(𝑥) 𝑑 𝑑𝑥

))⋅𝑥.

Let𝑙𝑜𝑝𝑖𝑙be the mould in𝐴𝑅𝐼_𝑎𝑟𝑖defined by the simple expression 𝑙𝑜𝑝𝑖𝑙(𝑣₁,…, 𝑣_𝑟) =𝑐_𝑟 𝑣₁+⋯+𝑣_𝑟

𝑣₁(𝑣₁−𝑣₂)⋯(𝑣_𝑟−1−𝑣_𝑟)𝑣_𝑟 (III.3.30) Set𝑝𝑖𝑙 =𝑒𝑥𝑝_𝑎𝑟𝑖(𝑙𝑜𝑝𝑖𝑙)where𝑒𝑥𝑝_𝑎𝑟𝑖denotes the exponential map associated to𝑝𝑟𝑒𝑎𝑟𝑖, and set𝑝𝑎𝑙 = 𝑠𝑤𝑎𝑝(𝑝𝑖𝑙).

The mould 𝑙𝑜𝑝𝑖𝑙 is easily seen to be both alternal and circ-neutral. It is also known (although surprisingly difficult to show) that the mould𝑙𝑜𝑝𝑎𝑙=𝑙𝑜𝑔_𝑎𝑟𝑖(𝑝𝑎𝑙)is alternal (cf. [Ec2], or [S2], Chap.

4.). Thus the moulds𝑝𝑖𝑙and𝑝𝑎𝑙 are both exponentials of alternal moulds and therefore symmetral.

The inverses of𝑝𝑎𝑙(in𝐺𝐴𝑅𝐼) and𝑝𝑖𝑙(in𝐺𝐴𝑅𝐼)are given by

𝑖𝑛𝑣𝑝𝑎𝑙=𝑒𝑥𝑝_𝑎𝑟𝑖(−𝑙𝑜𝑝𝑎𝑙), 𝑖𝑛𝑣𝑝𝑖𝑙=𝑒𝑥𝑝_𝑎𝑟𝑖(−𝑙𝑜𝑝𝑖𝑙).

The key maps we will be using in our proof are the adjoint operators associated to 𝑝𝑎𝑙 and𝑝𝑖𝑙, given by

𝐴𝑑_𝑎𝑟𝑖(𝑝𝑎𝑙) =𝑒𝑥𝑝(

𝑎𝑑_𝑎𝑟𝑖(𝑙𝑜𝑝𝑎𝑙))

, 𝐴𝑑_𝑎𝑟𝑖(𝑝𝑖𝑙) =𝑒𝑥𝑝(

𝑎𝑑_𝑎𝑟𝑖(𝑙𝑜𝑝𝑖𝑙))

, (III.3.31)

where𝑎𝑑_𝑎𝑟𝑖(𝑃)⋅𝑄=𝑎𝑟𝑖(𝑃 , 𝑄). The inverses of these adjoint actions are given by 𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙) =𝑒𝑥𝑝(

𝑎𝑑_𝑎𝑟𝑖(−𝑙𝑜𝑝𝑎𝑙))

, 𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙) =𝑒𝑥𝑝(

𝑎𝑑_𝑎𝑟𝑖(−𝑙𝑜𝑝𝑖𝑙))

. (III.3.32) These adjoint actions produce remarkable transformations of certain mould properties into others, and form the heart of much of Écalle’s theory of multizeta values. Écalle’sfundamental identityrelates the two adjoint actions of (III.3.31). Valid for all push-invariant moulds𝑀, it is given by

𝑠𝑤𝑎𝑝⋅𝐴𝑑_𝑎𝑟𝑖(𝑝𝑎𝑙)⋅𝑀 =𝑔𝑎𝑛𝑖𝑡(𝑝𝑖𝑐)⋅𝐴𝑑_𝑎𝑟𝑖(𝑝𝑖𝑙)⋅𝑠𝑤𝑎𝑝(𝑀), (III.3.33) where𝑝𝑖𝑐 ∈ 𝐺𝐴𝑅𝐼 is defined by𝑝𝑖𝑐(𝑣₁,…, 𝑣_𝑟) = 1∕𝑣₁⋯𝑣_𝑟 (see [Ec], or [S2], Theorem 4.5.2 for the complete proof).

For our purposes, it is useful to give a slightly modified version of this identity. Let𝑝𝑜𝑐 ∈𝐺𝐴𝑅𝐼 be the mould defined by

𝑝𝑜𝑐(𝑣₁,…, 𝑣_𝑟) = 1

𝑣₁(𝑣₁−𝑣₂)⋯(𝑣_𝑟−1−𝑣_𝑟). (III.3.34) Then𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)and𝑔𝑎𝑛𝑖𝑡(𝑝𝑖𝑐)are inverse automorphisms of𝐴𝑅𝐼_𝑙𝑢(see [B], Lemma 4.37). Thus, we can rewrite the above identity (III.3.33) as

𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑠𝑤𝑎𝑝⋅𝐴𝑑_𝑎𝑟𝑖(𝑝𝑎𝑙)⋅𝑀 =𝐴𝑑_𝑎𝑟𝑖(𝑝𝑖𝑙)⋅𝑠𝑤𝑎𝑝(𝑀), (III.3.35) and letting𝑁 =𝐴𝑑_𝑎𝑟𝑖(𝑝𝑎𝑙)⋅𝑀, i.e. 𝑀 =𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙)⋅𝑁, we rewrite it in terms of𝑁as

𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑠𝑤𝑎𝑝(𝑁) =𝑠𝑤𝑎𝑝⋅𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙)⋅𝑁 , (III.3.36) this identity being valid whenever𝑀 =𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙)⋅𝑁is push-invariant.

Step 3: Construction of the mapΞ

In this section we finally arrive at the main step of the construction of our map𝔨𝔯𝔳→𝔨𝔯𝔳_𝑒𝑙𝑙, namely the construction of the mapΞgiven in the following proposition.

Proposition III.3.20. The operatorΞ = 𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙)◦𝑝𝑎𝑟𝑖gives an injective Lie morphism of Lie subalgebras of𝐴𝑅𝐼_𝑎𝑟𝑖:

Ξ ∶𝐴𝑅𝐼^𝑝𝑜𝑙

𝑎𝑙+𝑠𝑒𝑛∗𝑐𝑖𝑟𝑐𝑐𝑜𝑛𝑠𝑡⟶𝐴𝑅𝐼𝑎𝑙+𝑝𝑢𝑠ℎ∗𝑐𝑖𝑟𝑐𝑛𝑒𝑢𝑡^Δ . (III.3.37) Proof. We have already shown that both spaces are Lie subalgebras of𝐴𝑅𝐼_𝑎𝑟𝑖, the first in Propo-sition III.3.17 and the second in III.3.2. Furthermore, since𝑝𝑎𝑟𝑖and𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙)are both invertible and respect the𝑎𝑟𝑖-bracket, the proposed map is indeed an injective map of Lie subalgebras. Thus it remains only to show that the image of𝐴𝑅𝐼𝑎𝑙+𝑠𝑒𝑛∗𝑐𝑖𝑟𝑐𝑐𝑜𝑛𝑠𝑡^𝑝𝑜𝑙 underΞreally lies in𝐴𝑅𝐼𝑎𝑙+𝑝𝑢𝑠ℎ∗𝑐𝑖𝑟𝑐𝑛𝑒𝑢𝑡^Δ . We will show separately that if𝐵∈𝐴𝑅𝐼𝑎𝑙+𝑠𝑒𝑛∗𝑐𝑖𝑟𝑐𝑐𝑜𝑛𝑠𝑡^𝑝𝑜𝑙 and𝐴= Ξ(𝐵), then

(i)𝐴is push-invariant, (ii)𝐴is alternal,

(iii)𝑠𝑤𝑎𝑝(𝐴)is circ-neutral up to addition of a constant-valued mould, (iv)𝐴∈𝐴𝑅𝐼^Δ.

Proof of (i). Écalle proved that 𝐴𝑑_𝑎𝑟𝑖(𝑝𝑎𝑙) transforms push-invariant moulds to moulds satisfying the senary relation (III.3.25) (see [Ec] (3.58); indeed this is how the senary relation arose). Since

𝐵 satisfies (III.3.23),𝐵̃ = 𝑝𝑎𝑟𝑖(𝐵) satisfies (III.3.25), so𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙)(𝐵) = Ξ(𝐵) =̃ 𝐴is push-invariant.

Proof of (ii). Recall that𝐴𝑅𝐼_𝑎𝑙is closed under𝑎𝑟𝑖and𝑒𝑥𝑝_𝑎𝑟𝑖(𝐴𝑅𝐼_𝑎𝑙)is the subgroup of symmetral moulds𝐺𝐴𝑅𝐼_{𝑔𝑎𝑟𝑖}^𝑎𝑠 of𝐺𝐴𝑅𝐼_{𝑔𝑎𝑟𝑖}.

The𝑝𝑎𝑙is known to be symmetral (cf. [Ec2], or in more detail [S2], Theorem 4.3.4).

Thus, since 𝐺𝐴𝑅𝐼_{𝑔𝑎𝑟𝑖}^𝑎𝑠 is a group, the𝑔𝑎𝑟𝑖-inverse mould 𝑖𝑛𝑣𝑝𝑎𝑙is also symmetral. Therefore the adjoint action 𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙) on 𝐴𝑅𝐼 restricts to an adjoint action on the Lie subalgebra𝐴𝑅𝐼_𝑎𝑙 of alternal moulds. If𝐵is alternal, then𝑝𝑎𝑟𝑖(𝐵)is alternal, and so𝐴= Ξ(𝐵)is alternal. This completes the proof of (ii).

For the assertions (iii) and (iv), we will make use of Écalle’s fundamental identity in the version (III.3.36) given in III.3.33, with𝑁 =𝑝𝑎𝑟𝑖(𝐵)(recall that (III.3.36) is valid whenever𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙)⋅𝑁 is push-invariant, which is the case for𝑝𝑎𝑟𝑖(𝐵)thanks to (i) above). The key point is that the operators 𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐) and𝐴𝑑_𝑎𝑟𝑖(𝑝𝑖𝑙)on the left-hand side of (III.3.36) are better adapted to tracking the circ-neutrality and the denominators than the right-hand operator𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙)considered directly.

Proof of (iii). Let𝑏 ∈ 𝑊_𝔨𝔯𝔳, and assume that𝑏is of homogeneous degree𝑛. Let𝐵 = 𝑚𝑎(𝑏). Then by Proposition III.3.15 and Proposition II.3.22,𝑠𝑤𝑎𝑝(𝐵)is circ-constant, and even circ-neutral if𝑛is even.

We need to show that𝑠𝑤𝑎𝑝⋅𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙)⋅𝑝𝑎𝑟𝑖(𝐵)is *circ-neutral. To do this, we use (III.3.36) with𝑁 =𝑝𝑎𝑟𝑖(𝐵), and in fact show the result on the left-hand side, which is equal to

𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑝𝑎𝑟𝑖⋅𝑠𝑤𝑎𝑝(𝐵)

(noting that𝑝𝑎𝑟𝑖commutes with𝑠𝑤𝑎𝑝). We prove that this mould is *circ-neutral in three steps. First we show that the operator𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑝𝑎𝑟𝑖changes a circ-constant mould into one that is circ-neutral (Proposition III.3.21). Secondly, we show that the operator𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)preserves the property of circ-neutrality (Proposition III.3.23). Finally, we show that if𝑀 is a mould that is not circ-constant but only *circ-constant, and if𝑀₀is the (unique) constant-valued mould such that𝑀 +𝑀₀ is circ-constant, then

𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑝𝑎𝑟𝑖(𝑀) +𝑀₀

is circ-neutral. Using (III.3.36), this will show that𝑠𝑤𝑎𝑝⋅𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙)⋅𝑀 is *circ-neutral.

Proposition III.3.21. Fix𝑛 ≥ 3, and let𝑀 ∈ 𝐴𝑅𝐼 be a circ-constant polynomial-valued mould of homogeneous degree𝑛. Then𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑝𝑎𝑟𝑖(𝑀)is circ-neutral.

Proof. Let 𝑐 = (

𝑀(𝑣₁)|𝑣^𝑛−1₁ )

, and let 𝑁 = 𝑝𝑎𝑟𝑖(𝑀), so that 𝑁(𝑣₁) = −𝑐𝑣^𝑛−1₁ . Let 𝐯 = (𝑣₁,…, 𝑣_𝑟), and let𝐖_𝐯 be the set of decompositions𝑑_𝐯 of𝐯 into chunks 𝑑_𝐯 = 𝐚₁𝐛₁⋯𝐚_𝑠𝐛_𝑠 as in (III.3.27). For any decomposition𝑑_𝐯, we let its𝐛-part be the unordered set{𝐛₁,…,𝐛_𝑠}, its𝐚-part the unordered set{𝐚₁,…,𝐚_𝑠}, and we write𝑙_𝐚for the number of letters in the𝐚-part, i.e.𝑙_𝐚 =|𝐚₁|+⋯+

|𝐚_𝑠|.

Let

𝐖=∐

𝑖

𝐖_𝜎𝑖 𝑟(𝐯),

where the 𝜎_𝑟^𝑖(𝐯) are the cyclic permutations of 𝐯 = (𝑣₁,…, 𝑣_𝑟), and let 𝐖^𝐛 denote the subset of decompositions in𝐖having identical𝐛-part. The decompositions in𝐖having identical𝐛-part to a given decomposition𝑑_𝐯∈𝐖_𝐯are as follows: there is exactly one decomposition in𝐖_𝜎𝑖−1

𝑟 (𝐯)for each 𝑖such that𝑣_𝑖 is one of the letters in the𝐚-part of 𝐯, which is obtained from𝑑_𝐯 by placing dividers between the same letters. For example, if𝑟 = 5and𝑑_𝐯 = 𝐚₁𝐛₁𝐚₂𝐛₂ = (𝑣₁, 𝑣₂)(𝑣₃)(𝑣₄)(𝑣₅)then the

two other decompositions having the same𝐛-part{(𝑣₃),(𝑣₅)} are given by(𝑣₂)(𝑣₃)(𝑣₄)(𝑣₅)(𝑣₁)and (𝑣₄)(𝑣₅)(𝑣₁, 𝑣₂)(𝑣₃). Thus if𝐛 denotes the 𝐛-part of a given decomposition𝑑_𝐯 of𝐯 = (𝑣₁,…, 𝑣_𝑟), then𝐖^𝐛contains exactly𝑙_𝐚decompositions, more precisely exactly one decomposition of each cyclic permutation(𝑣_𝑖,…, 𝑣_𝑟, 𝑣₁,…, 𝑣_𝑖−1)with𝑣_𝑖in the𝐚-part of𝑑_𝐯. Since𝑀 is circ-constant for𝑐, we have

𝑁(𝑣₁,…, 𝑣_𝑟) +⋯+𝑁(𝑣_𝑟, 𝑣₁, ..., 𝑣_𝑟−1) = (−1)^𝑟𝑐∑

𝑤

𝑤. (III.3.38)

By the explicit formula (III.3.28), we have (𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑁)

(𝑣₁,…, 𝑣_𝑟) =∑

𝐖_𝐯

𝑝𝑜𝑐(⌊𝐛₁)⋯𝑝𝑜𝑐(⌊𝐛_𝑠)𝑁(𝐚₁⋯𝐚_𝑠), (III.3.39) so adding up over the cyclic permutations of𝐯, we have

∑𝑟−1 where the last equality follows from (III.3.38).

If𝑐 = 0, the expression (III.3.40) is trivially equal to zero in all depths𝑟 > 1, so we obtained the desired result that𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑝𝑎𝑟𝑖(𝑀)is circ-neutral. In order to deal with the case where𝑀 is circ-constant for a value𝑐≠0, we use a trick and subtract off a known mould that is also circ-constant for𝑐.

Lemma III.3.22. For 𝑛 > 1and any constant𝑐, let𝑇_𝑐^𝑛 be the homogeneous polynomial mould of degree𝑛defined by

𝑇_𝑐^𝑛(𝑣₁,…, 𝑣_𝑟) = 𝑐 𝑟𝑃_𝑟^𝑛,

where𝑃_𝑛^𝑟 is the sum is over all monomials of degree𝑛−𝑟in the variables𝑣₁,…, 𝑣_𝑟for1≤ 𝑟 ≤ 𝑛.

Then𝑇_𝑐^𝑛is circ-constant and𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑝𝑎𝑟𝑖(𝑇_𝑐^𝑛)is circ-neutral.

The proof of this lemma is annoyingly technical, so we have relegated it to Appendix 2. Consider the mould𝑁 = 𝑀 −𝑇_𝑐^𝑛. The mould𝑁 is circ-constant since𝑀 and𝑇_𝑐^𝑛both are, but𝑁(𝑣₁) = 0, so by the result above, we know that𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑝𝑎𝑟𝑖(𝑁)is circ-neutral. But Lemma III.3.22 shows that𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑝𝑎𝑟𝑖(𝑇_𝑐^𝑛)is circ-neutral, so the mould𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑝𝑎𝑟𝑖(𝑀) is also circ-neutral, as desired.

We now proceed to the second step, showing that the operator𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)preserves circ-neutrality.

Proposition III.3.23. If𝑀 ∈𝐴𝑅𝐼is circ-neutral then𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑀 is also circ-neutral.

Proof. By (III.3.32), we have

𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙) =𝑒𝑥𝑝(

𝑎𝑑_𝑎𝑟𝑖(−𝑙𝑜𝑝𝑖𝑙))

=∑

𝑛≥0

(−1)^𝑛

𝑛 𝑎𝑑_𝑎𝑟𝑖(𝑙𝑜𝑝𝑖𝑙)^𝑛. (III.3.41) The definition of𝑙𝑜𝑝𝑖𝑙 in (III.3.30) shows that𝑙𝑜𝑝𝑖𝑙is trivially circ-neutral. Thus, since𝑀 is circ-neutral,𝑎𝑑_𝑎𝑟𝑖(𝑙𝑜𝑝𝑖𝑙)⋅𝑀 = 𝑎𝑟𝑖(𝑙𝑜𝑝𝑖𝑙, 𝑀)is also circ-neutral by Lemma III.3.10, and successively so are all the terms𝑎𝑑_𝑎𝑟𝑖(𝑙𝑜𝑝𝑖𝑙)^𝑛(𝑀). Thus𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑀 is circ-neutral.

Finally, we now assume that 𝑠𝑤𝑎𝑝(𝐵) is a *circ-neutral polynomial-valued mould in 𝐴𝑅𝐼 of homogeneous degree𝑛. Let 𝐵₀ be the (unique) constant-valued mould such that 𝑠𝑤𝑎𝑝(𝐵) + 𝐵₀ is circ-neutral. Then by Propositions III.3.21 and III.3.23, the mould

𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑝𝑎𝑟𝑖(𝐵+𝐵₀) is circ-neutral. This mould breaks up as the sum

𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑝𝑎𝑟𝑖(𝐵) +𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑝𝑎𝑟𝑖(𝐵₀),

but the operator𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)preserves constant-valued moulds (cf. [S], Lemma 4.6.2 for the proof). Thus

𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑝𝑎𝑟𝑖(𝐵+𝐵₀) =𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑝𝑎𝑟𝑖(𝐵) +𝐵₀, so

𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑝𝑎𝑟𝑖(𝐵) =𝑠𝑤𝑎𝑝⋅Ξ(𝐵) is *circ-neutral, completing the proof of (iii).

Proof of (iv).We will again use the left-hand side of (III.3.36), this time to track the denominators that appear in the right-hand side. By (III.3.36), if𝐵 is a polynomial-valued mould satisfying the senary relation, and if𝐴= Ξ(𝐵) =𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙)⋅𝑝𝑎𝑟𝑖(𝐵), then𝐴lies in𝐴𝑅𝐼^Δif and only if

𝑠𝑤𝑎𝑝⋅𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑠𝑤𝑎𝑝(

𝑝𝑎𝑟𝑖(𝐵))

∈𝐴𝑅𝐼^Δ. (III.3.42) We will prove that this is the case, by studying the denominators that are produced, first by applying 𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)to a polynomial-valued mould, and then by applying𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙). The first result is that the denominators introduced by applying𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)are at worst of the form(𝑣₁−𝑣₂)⋯(𝑣_𝑟−1−𝑣_𝑟). Lemma III.3.24. Let𝑀 ∈𝐴𝑅𝐼^𝑝𝑜𝑙. Then

𝑠𝑤𝑎𝑝⋅𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑀 ∈𝐴𝑅𝐼^Δ.

Proof. The explicit expression for𝑔𝑎𝑛𝑖𝑡(𝑄)given in (III.3.28) shows that the only denominators that can occur in𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑀 come from the factors

𝑝𝑜𝑐(⌊𝐛₁)⋯𝑝𝑜𝑐(⌊𝐛_𝑠) (III.3.43)

for all decompositions𝑑_𝐯=𝐚₁𝐛₁⋯𝐚_𝑠𝐛₂of𝐯= (𝑣₁,…, 𝑣_𝑟)into chunks as in (III.3.27), and

⌊𝐛_𝑖= (𝑣_𝑘−𝑣_𝑘−1, 𝑣_𝑘+1−𝑣_𝑘−1,…, 𝑣_𝑘+𝑙−𝑣_𝑘−1)

(for𝑘 > 1) as in (III.3.29). Since𝑝𝑜𝑐 is defined as in (III.3.34), the only factors that can appear in (III.3.43) are(𝑣_𝑙−𝑣_𝑙−1)where𝑣_𝑙is a letter in one of𝐛_𝑖, and these factors appear in each term with multiplicity one. Since the sum ranges over all possible decompositions, the only letter of𝐯that never belongs to any𝐛_𝑖is𝑣₁; all the other factors(𝑣_𝑖−𝑣_𝑖−1)appear. Thus(𝑣₁−𝑣₂)(𝑣₂−𝑣₃)⋯(𝑣_𝑟−1−𝑣_𝑟)is a common denominator for all the terms in the sum defining𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑀. The swap of this common denominator is equal to𝑢₂⋯𝑢_𝑟, so this term is a common denominator for𝑠𝑤𝑎𝑝⋅𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑀, which proves the lemma.

Lemma III.3.25. Let𝑀 , 𝑁 ∈ 𝐴𝑅𝐼_{∗𝑐𝑖𝑟𝑐𝑛𝑒𝑢𝑡}be two moulds such that𝑠𝑤𝑎𝑝(𝑀)and𝑠𝑤𝑎𝑝(𝑁)lie in 𝐴𝑅𝐼^Δ. Then𝑠𝑤𝑎𝑝(

𝑎𝑟𝑖(𝑀 , 𝑁))

also lies in𝐴𝑅𝐼^Δ.

Proof.In Proposition A.1 of the Appendix of [BS], it is shown that if𝑀and𝑁are alternal moulds in𝐴𝑅𝐼such that𝑠𝑤𝑎𝑝(𝑀)and𝑠𝑤𝑎𝑝(𝑁)lie in𝐴𝑅𝐼^Δ, then𝑠𝑤𝑎𝑝(

𝑎𝑟𝑖(𝑀 , 𝑁))

also lies in𝐴𝑅𝐼^Δ. In fact, it is shown in Proposition A.2 of that appendix that alternal moulds𝑀 whose swap lies in𝐴𝑅𝐼^Δ satisfy the following property: setting

𝑀̌(𝑣₁,…, 𝑣_𝑟) =𝑣₁(𝑣₁−𝑣₂)⋯(𝑣_𝑟−1−𝑣_𝑟)𝑣_𝑟𝑀(𝑣₁,…, 𝑣_𝑟), we have

𝑀̌(0, 𝑣₂,…, 𝑣_𝑟) =𝑀̌ (𝑣₂,…, 𝑣_𝑟,0). (III.3.44) In fact, the proof that𝑠𝑤𝑎𝑝(

𝑎𝑟𝑖(𝑀 , 𝑁))

lies in𝐴𝑅𝐼^Δdoes not use the full alternality of𝑀 and𝑁, but only (III.3.44). Therefore, the same proof goes through when𝑀 and𝑁are *circ-neutral moulds such that𝑠𝑤𝑎𝑝(𝑀)and𝑠𝑤𝑎𝑝(𝑁)lie in𝐴𝑅𝐼^Δ, as long as we check that every *circ-neutral mould𝑀 such that𝑠𝑤𝑎𝑝(𝑀) ∈𝐴𝑅𝐼^Δsatisfies (III.3.44).

To check this, let𝑀 be such a mould; by additivity, we may assume that𝑀 is concentrated in a single depth𝑟 >1. This means that there is a constant𝐶_𝑀 such that

𝑀(𝑣₁,…, 𝑣_𝑟) +𝑀(𝑣₂,…, 𝑣_𝑟, 𝑣₁) +⋯+𝑀(𝑣_𝑟, 𝑣₁,…, 𝑣_𝑟−1) =𝐶_𝑀, which we can also write as

𝑀̌(𝑣₁,…, 𝑣_𝑟)

𝑣₁(𝑣₁−𝑣₂)⋯(𝑣_𝑟−1−𝑣_𝑟)𝑣_𝑟 +

𝑀̌(𝑣₂,…, 𝑣_𝑟, 𝑣₁)

𝑣₂(𝑣₂−𝑣₃)⋯(𝑣_𝑟−1−𝑣_𝑟)(𝑣_𝑟−𝑣₁)𝑣₁+

⋯+

𝑀̌(𝑣_𝑟, 𝑣₁,…, 𝑣_𝑟−1)

𝑣_𝑟(𝑣_𝑟−𝑣₁)⋯(𝑣_𝑟−2−𝑣_𝑟−1)𝑣_𝑟−1 =𝐶_𝑀

where the numerators are polynomials. If we multiply the entire equality by𝑣₁and set𝑣₁ = 0, only the first two terms do not vanish, and they yield precisely the desired relation (III.3.44).

Corollary III.3.26. If𝑁 ∈𝐴𝑅𝐼 is a *circ-neutral mould such that𝑠𝑤𝑎𝑝(𝑁) ∈𝐴𝑅𝐼^Δ, then also 𝑠𝑤𝑎𝑝⋅𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑁 ∈𝐴𝑅𝐼^Δ. (III.3.45) Proof. The lemma shows that𝑠𝑤𝑎𝑝⋅𝑎𝑟𝑖(𝑙𝑜𝑝𝑖𝑙, 𝑁) ∈ 𝐴𝑅𝐼^Δsince the mould𝑙𝑜𝑝𝑖𝑙is circ-neutral and𝑠𝑤𝑎𝑝⋅𝑙𝑜𝑝𝑖𝑙 ∈ 𝐴𝑅𝐼^Δby (III.3.30). In fact, applying the lemma successively shows that𝑠𝑤𝑎𝑝⋅ 𝑎𝑑_𝑎𝑟𝑖(𝑙𝑜𝑝𝑖𝑙)^𝑛(𝑁) ∈𝐴𝑅𝐼^Δfor all𝑛≥1. Since𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑁 is obtained by summing these terms by (III.3.41), we obtain (III.3.45).

To conclude, we set𝑀 =𝑠𝑤𝑎𝑝⋅𝑝𝑎𝑟𝑖(𝐵); then by Lemma III.3.24 we have 𝑠𝑤𝑎𝑝⋅𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑠𝑤𝑎𝑝⋅𝑝𝑎𝑟𝑖(𝐵) ∈𝐴𝑅𝐼^Δ.

By Proposition III.3.21 this mould is *circ-neutral, so we can apply Corollary III.3.26 with 𝑁 = 𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑠𝑤𝑎𝑝⋅𝑝𝑎𝑟𝑖(𝐵)to conclude that

𝑠𝑤𝑎𝑝⋅𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑖𝑙)⋅𝑔𝑎𝑛𝑖𝑡(𝑝𝑜𝑐)⋅𝑠𝑤𝑎𝑝⋅𝑝𝑎𝑟𝑖(𝐵) ∈𝐴𝑅𝐼^Δ. Thus thus by (III.3.36) with𝑁 =𝑝𝑎𝑟𝑖(𝐵), we finally find that

𝐴𝑑_𝑎𝑟𝑖(𝑖𝑛𝑣𝑝𝑎𝑙)⋅𝑝𝑎𝑟𝑖(𝐵) = Ξ(𝐵) ∈𝐴𝑅𝐼^Δ, which completes the proof of (iv).

We have thus finished proving Proposition III.3.20. Backtracking, this means we have completed the details of Step 3 of the proof of Theorem III.3.12. Step 4, the final step in the proof, is very easy and was explained completely just before paragraph III.3.3. Thus we have now completed the proof of Theorem III.3.12, i.e. we have completed the construction of the injective Lie algebra morphism 𝔨𝔯𝔳↪𝔨𝔯𝔳_𝑒𝑙𝑙.

III.3.4 Relations with elliptic Grothendieck-Teichmüller and double shuffle

Dans le document A mould-theoretic perspective on Kashiwara-Vergne theory (Page 79-90)