Elise Raphael

Leila Schneps

On linearised and elliptic versions of the Kashiwara-Vergne Lie algebra

The goal of this article is to define a linearized or depth-graded version lkv, and a closely related elliptic version krv ell , of the Kashiwara-Vergne Lie algebra krv originally constructed by Alekseev and Torossian as the space of solutions to the linearized Kashiwara-Vergne problem. We show how the elliptic Lie algebra krv ell is related to earlier constructions of elliptic versions grt ell and ds ell of the Grothendieck-Teichmüller Lie algebra grt and the double shuffle Lie algebra ds.

In particular we show that there is an injective Lie morphism ds ell → krv ell , and an injective Lie algebra morphism krv → krv ell extending the known morphisms grt → grt ell (Enriquez section) and ds → ds ell ( Écalle map).

Introduction

This article studies two Lie algebras closely related to the Kashiwara-Vergne Lie algebra krv defined in [AT]: firstly, a linearized (or depth-graded) version lkv, and secondly, an elliptic version krv ell whose construction is closely related to that of lkv. The results are motivated by the comparison of krv with two other Lie algebras familiar from the theory of multiple zeta values: the Grothendieck-Teichmüller Lie algebra grt and the double shuffle Lie algebra ds. Our definition of lkv is an analog of the definition of the bigraded linearized double shuffle Lie algebra ls, whose structure has given rise to many results and conjectures, in particular the famous Broadhurst-Kreimer conjecture. Our definition of krv ell is an analog of the definition of the elliptic double shuffle Lie algebra ds ell , which itself is related on the one hand to ls and on the other to the elliptic Grothendieck-Teichmüller Lie algebra grt ell . We explore all the relations between these different objects.

Like grt and ds, the Lie algebra krv is equipped with a depth filtration; we write gr for the associated gradeds. We show that in analogy with the known injective map gr ds → ls, there is an injective map gr krv → lkv (Proposition 2). We also show that the injective map gr ds → gr krv arising from the injective Lie morphism ds → krv extends to an injective Lie morphism ls → lkv, and that the parts of these spaces of depths d = 1, 2, 3 are isomorphic for all weights n (Theorem 3), which yields the dimensions of the bigraded parts of lkv and gr krv of depths 1, 2, 3 in all weights, since these dimensions are well-known for ls. Finally, we define the elliptic version krv ell as a subspace of derivations of the free Lie algebra on two generators, and prove that it is closed under the Lie bracket of derivations (Theorem 5). We also define an injective Lie morphism krv → krv ell (Theorem 6) in analogy with the section map grt → grt ell ( [E]) and the mould-theoretic double shuffle map ds → ds ell ([S3]). Finally, although we were not able to prove the existence of an injection grt ell → krv ell , we define a Lie subalgebra grt ell ⊂ grt ell such that the following diagram commutes:

grt → ds → krv ↓ ↓ ↓ grt ell → ds ell → krv ell 1
The main technique used for the constructions in this article is the mould theory developed by J. Écalle, to which we provide a brief introduction in §3, with complements in §4.

1.1. Special types of derivations of lie 2 . Let lie 2 denote the degree completion of the free Lie algebra over Q on non-commutative variables x and y. The Lie algebra lie 2 has a weight grading by the degree (=weight) of the polynomials, and a depth grading by the y-degree (=depth) of the polynomials. We write (lie 2 ) n for the graded part of weight n, (lie 2 ) r for the graded part of depth r, and (lie 2 ) r n for the intersection, which is finite-dimensional.

All the Lie algebras we will study in this article (the well-known ones krv, grt and ds as well as the linearized ls, and the spaces lkv and krv ell that we introduce) can be viewed either as Lie subalgebras of particular subalgebras of the derivations of lie 2 , equipped with the bracket of derivations, or as subspaces of lie 2 equipped with particular Lie brackets coming from the Lie bracket of derivations. Both ways of considering our spaces are natural and useful, and we go back and forth between them as convenient for our proofs.

Let der 2 denote the algebra of derivations on lie 2 . It is a Lie algebra under the Lie bracket given by the commutator of derivations. For a, b ∈ lie 2 , we write D b,a for the derivation defined by x → b and y → a. The bracket is explicitly given by • Let oder 2 denote the Lie subalgebra of der 2 of derivations D = D b,a that annihilate the bracket [x, y] and such that neither D(x) nor D(y) have a linear term in x. The map oder 2 → lie 2 given by D → D(x) is injective (see Corollary 18).

• Let tder 2 denote the Lie subalgebra of der 2 of tangential derivations, which are the derivations E a,b for elements a, b ∈ lie 2 such that a has no linear term in x and b has no linear term in y, such that there exists c ∈ lie 2 such that setting z = -x -y, i.e. the second term of (4).

We have the following diagram showing the connections between these subspaces:

(7)

oder 2 → der 2 ↑ tder 2 / \ sder 2 ∼ → ider 2
The isomorphism between sder 2 and ider 2 is given in Lemma 25.

1.2. Definition of the Kashiwara-Vergne Lie algebra krv. The universal enveloping algebra of lie 2 is isomorphic to the degree completion Ass 2 = Q x, y of the free associative algebra with non-commutative generators x, y, i.e. to the ring of power series in x and y.

Definition 1. The trace vector space tr 2 (cf. [AT]) is defined to be the quotient of Ass 2 by the equivalence relation given between words in x and y by w ∼ w if w can be obtained from w by a cyclic permutation of the letters of the word w, and extended linearly to polynomials. The natural projection is denoted tr : Ass 2 → tr 2 .

For any polynomial f ∈ Ass 2 with constant term c, we can decompose f in two ways as (8) f = c + f x x + f y y = c + xf x + yf y for uniquely determined polynomials f x , f y , f x , f y in Ass 2 .

Definition 2. The divergence map is given by div : tder 2 -→ tr 2 u = E a,b -→ tr(a x x + b y y).

Definition 3. The Kashiwara-Vergne Lie algebra krv 2 is defined to be the subspace of sder 2 of derivations E a,b such that there exists a one-variable power series h(x) ∈ Q[x] of degree ≥ 2 such that (9) div (E a,b ) = tr h(x + y) -h(x) -h(y) .

This definition comes from [AT], where it was shown that krv 2 is actually a Lie subalgebra of sder 2 . This Lie algebra inherits a weight-grading from that of lie 2 , for which E a,b is of weight n if b (and thus also a) is a Lie polynomial of homogeneous degree n. In particular, the weight 1 part of krv 2 is spanned by the single element u = E y,x , and the weight 2 part is zero. In this article, we do not consider the weight 1 part of krv 2 . For convenience, we set krv = ⊕ n≥3 (krv 2 ) n , where (krv 2 ) n denotes the weight graded part of krv 2 of weight n. We have

krv 2 = (krv 2 ) 1 ⊕ krv = Q[E y,x ] ⊕ krv.
Because the other Lie algebras in the literature that are most often compared with the Kashiwara-Vergne Lie algebra have no weight 1 or weight 2 parts, it makes most sense to compare them with krv. Thus it is krv that we study for the remainder of this article.

The Lie algebra krv also inherits a depth filtration from the depth grading on lie 2 , for which E a,b is of depth r if r is the smallest number of y's occurring in any monomial of b. We write gr krv for the associated graded for this depth filtration, so that gr krv is a Lie algebra that is bigraded for the weight and the depth; we write gr r n krv for the part of weight n and depth r. Essentially, an element of gr krv is a derivation [x, [x, [x, y]]]].

ā = [x, [x, [[x, y], y]]] -2[[x, [x, y]], [x, y]], b = [x,
1.3. The Grothendieck-Teichmüller and double shuffle Lie algebras. Recall that the Grothendieck-Teichmüller Lie algebra grt is the space of polynomials b ∈ lie 2 satisfying the famous pentagon relation, equipped with the Poisson bracket (6). This algebra was first introduced by Y. Ihara in [I], with three defining relations, as a particular derivation algebra of lie 2 (via the association b → d b as in ( 5)); H. Furusho subsequently showed that the pentagonal relation implies the other two (cf. [F1]).

Recall also that the double shuffle Lie algebra ds is the space of polynomials b ∈ lie 2 satisfying a particular set of conditions on the coefficients called the stuffle relations, studied in the first place by Racinet (cf. [R]), who gave a quite difficult proof that ds is also a Lie algebra under the Poisson bracket (6). This proof was later somewhat streamlined by Furusho (cf. [F2], Appendix), and a recent preprint [EF] gives another proof with a different approach, identifying the space as a stabilizer. Putting together basic elements from Écalle's mould theory also yields a completely different and very simple proof of this result ( [SS]).

There is a commutative triangle of injective Lie morphisms grt → ds

(10) \/ krv.
The existence of the injection grt → ds was proven in [F1]; it is given by b(x, y) → b(x, -y). The existence of the injection grt → krv was proven in [AT]; it is given by b(x, y) → b(z, y) where z = -x -y. Finally, the existence of the injection ds → krv was proven in [S1] (using results from Écalle's mould theory), and is given, of course, by b(x, y) → b(z, -y). In particular, these morphisms respect the weight gradings and depth filtrations on all three spaces.

1.4. The linearized Kashiwara-Vergne Lie algebra: main results. For i ≥ 1, set C i = ad(x)

lie 2 Q[x] ⊕ lie C .
Thus, Lazard elimination shows that every polynomial b ∈ lie 2 having no linear term in x can be written uniquely as a Lie polynomial in the C i .

Definition 4. Let the push-operator be defined on monomials in x, y by ( 12)

push(x a0 yx a1 y • • • yx ar ) = x ar yx a0 y • • • yx ar-1 .
The push is considered to act trivially on constants and powers of x n , so we can extend it to all of Ass 2 by linearity. A polynomial b in x, y is said to be • push-invariant if push(b) = b, and

• push-neutral if b r + push(b r ) + • • • + push r (b r ) = 0 for all r ≥ 1, where b r denotes the depth r part of b. Finally, we say that b is • circ-neutral if b y is push-neutral in depths r > 1.
Definition 5. The linearized Kashiwara-Vergne Lie algebra lkv is the space of elements b ∈ lie C of degree ≥ 3 such that (i) b is push-invariant, and (ii) b is circ-neutral.

Our first result on lkv is that is a bigraded Lie algebra.

Proposition 1. The space lkv is bigraded by weight and depth, and forms a Lie algebra under the Poisson bracket defined in (6).

In §1.4 below, we define a larger space, the elliptic Kashiwara-Vergne Lie algebra krv ell , and show in Theorem 5 that it is a Lie algebra. Although it might be possible (albeit laborious) to prove Proposition 1 directly, it turns out to follow immediately from Theorem 5, due to the fact that there is a simple injection of lkv into the larger space krv ell (see Proposition 6 following Theorem 5) whose image is easily identifiable as the intersection of two Lie subalgebras. For this reason, the proof of Proposition 1 can be found in Corollary 22 at the end of §4.1, following the proof of Theorem 5.

In §2, we show how we derive the definition of lkv via a reformulation of the defining properties of krv, in the sense that the defining properties of lkv are merely truncations of the two reformulated defining properties of krv to their lowest-depth parts. This construction automatically gives the following result on lkv, whose proof is in §2.3. Proposition 2. There is an injective Lie algebra morphism gr krv → lkv.

We conjecture that these two spaces are in fact isomorphic.

In using this type of definition for lkv, we are following the analogous situation of the well-known double shuffle Lie algebra ds and the associated linearized double shuffle space ls studied in many articles (cf. for example [Br]). The bigraded linearized space ls is defined as the set of Lie polynomials f ∈ lie 2 of weight n ≥ 3 such that the polynomial f y y, rewritten in the variables y i = x n-1 y for n ≥ 1, is an element of the free Lie algebra on the y i . One also adds the extra assumption that if f is of depth 1, then it is of odd weight, an assumption which is not needed for lkv as it follows from the push-invariance condition in the definition. By its very construction, there is an injective Lie algebra homomorphism (13) gr ds → ls, and it is conjectured that these two spaces are isomorphic, but like for lkv, this is still an open question.

The injective Lie algebra morphism (10) from ds to krv yields a corresponding bigraded injective map: (14) gr ds → gr krv.

Our next result extends this map to the more general linearized spaces ls and lkv.

Theorem 3. The Lie injection ( 14) extends to a bigraded Lie injection on the associated linearized spaces, giving the following commutative diagram:

gr ds → gr krv. ↓ ↓ ls → lkv.
For all n ≥ 3 and r = 1, 2, 3, the map is an isomorphism of the bigraded parts

ls r n lkv r n .
This theorem will be proved in §3, using mould theory, to which we give a brief and elementary introduction. Mould theory is also essential for all the proofs concerning the elliptic Kashiwara-Vergne Lie algebra defined in the next subsection.

Adding a variety of known results in the depth 2 and depth 3 situations to this result, we obtain the following corollary.

Corollary 4. The following spaces are isomorphic for n ≥ 3 and r = 1, 2, 3: gr r n grt gr r n ds gr r n krv ls r n lkv r n . In particular, all of these spaces are zero when r = 1 or 3 and n is even, or when r = 2 and n is odd.

Proof. The dimensions of the spaces gr r n grt, gr r n ds and ls r n in depths are known to be equal to each other in depths r ≤ 3 ( [R], [G]). Since the injective map ds → krv (10) induces a map gr ds → gr krv, Proposition 2 shows that gr r n krv is sandwiched between gr r n ds and lkv r n ; by the theorem, all five spaces are then equal when r ≤ 3. (The dimensions of gr 2 n krv were also computed in [ALR], without using the comparison with double shuffle.)

We conjecture that lkv r n ls r n for all n, r, and calculations up to about n = 15 bear this conjecture out, but we were not able to prove the isomorphism for any other cases, not even the special case n ≡ r mod 2, where it is well-known that gr r n grt = gr r n ds = gr r n krv = ls r n = 0 (cf.

[IKZ], [Br] for classical proofs, or [S2] for the exposition of Écalle's mould-theoretic proof).

Let us end this subsection by giving the mould-language reformulation of the definition of lkv, which will allow us to connect it directly to the definition of the elliptic Kashiwara-Vergne Lie algebra defined in the next subsection, which cannot be defined directly in terms of elements of lie 2 . The mould definition of lkv clearly echoes the definition in terms of Lie elements given above; the equivalence is shown in detail in §3.

Definition 5 : Mould-reformulated lkv. The linearized Kashiwara-Vergne Lie algebra lkv is the space of elements b ∈ lie C of degree ≥ 3 such that writing the depth r part of b as

(15) b r = a k a C a1 • • • C ar
where the sum runs over tuples a = (a 1 , . . . , a r ), a i ≥ 1, and setting

(16) B r (u 1 , . . . , u r ) = a k a u a1-1 1 • • • u ar-1 r
for commutative variables u 1 , . . . , u r and ( 17)

Br (v 1 , . . . , v r ) = B r (v r , v r-1 -v r , . . . , v 1 -v 2 )
for commutative variables v 1 , . . . , v r , we have the two properties: (i) B r is push-invariant for r ≥ 1, i.e.

(18) B(u 0 , u 1 , . . . , u r-1 ) = B(u 1 , . . . , u r )

where

u 0 = -u 1 -• • • -u r , and
(ii) Br is circ-neutral for r > 1, i.e.

(19) Br (v 1 , . . . , v r ) + Br (v 2 , . . . , v r , v 1 ) + • • • + Br (v r , v 1 , . . . , v r-1 ) = 0.

1.5. The elliptic Kashiwara-Vergne Lie algebra. The last section of this article is devoted to the study of the elliptic Kashiwara-Vergne Lie algebra. The definition of this algebra is based on that of the linearized Lie algebra lkv, differing only from Definition 5 by the denominator appearing in (20), which makes it impossible to express it directly in terms of Lie elements like Definition 5.

Definition 6. The elliptic Kashiwara-Vergne vector space krv ell is spanned by the elements b ∈ lie C such that writing the depth r part b r as in (15) and the associated polynomial B r as in ( 16), and setting

(20) B r * (u 1 , . . . , u r ) = 1 u 1 • • • u r (u 1 + • • • + u r ) B r (u 1 , . . . , u r ) and Br * = B r * (v r , v r-1 -v r , . . . , v 1 -v 2 ), we have (i) B r
* is push-invariant as in (18) for r ≥ 1; (ii) Br

* is circ-neutral as in (19) for r > 1.

The first main result on krv ell is of course that it is a bigraded Lie algebra, but this comes from an injective map from krv ell into oder 2 rather than into sder 2 as for lkv.

Theorem 5. (i) The space krv ell is bigraded for the weight and the depth. (ii) For each b ∈ krv ell , there exists a unique polynomial a ∈ lie C , called the partner of b, such that D b,a ∈ oder 2 . (iii) The image of the injective linear map b → D b,a is a Lie subalgebra of oder 2 ; in other words krv ell is a Lie algebra under the Lie bracket coming from the bracket of derivations as in ( 1) and ( 2).

This theorem is proven in §4.1 (Theorem 19); it necessitates the introduction of some more complicated definitions and results from mould theory than those used in §3.

The following result is key to the comparison of lkv and krv ell , and to the proof that lkv is a Lie algebra. Proposition 6. There is an injective linear map

lkv → krv ell b(x, y) → [x, b(x, [x, y])] (22)
whose image is a Lie subalgebra of krv ell . Equivalently, the map can be defined on the family B r of polynomials in commutative variables associated to b as in ( 16) by

B r (u 1 , . . . , u r ) → u 1 • • • u r (u 1 + • • • + u r )B r (u 1 , . . . , u r ).
This reflects the fact that by Definition 6, krv ell is isomorphic to the space spanned by the polynomials in the commutative variables u i that become pushinvariant and circ-neutral (possibly after adding a constant) after division by u 1 • • • u r (u 1 + • • • + u r ), while lkv is isomorphic to space of polynomials that are themselves pushinvariant and circ-neutral: thus multiplying by the factor u 1 • • • u r (u 1 + • • • + u r ) maps lkv precisely to the subspace of krv ell consisting of polynomials that remain polynomial after division by

u 1 • • • u r (u 1 + • • • + u r ).
In two independent articles, H. Tsunogai [Ts] and B. Enriquez [E] defined a Lie algebra that Enriquez calls the elliptic Grothendieck-Teichmüller Lie algebra grt ell , based on the idea that just as Ihara had defined grt as the algebra of derivations on lie 2 (identified with the braid Lie algebra on four strands) that extend to a particular type of derivation on the braid Lie algebra on five strands, grt ell is the Lie algebra of derivations on lie 2 (now identified with the genus one braid Lie algebra on two strands) that extend to a very particular type of derivation of the genus one braid Lie algebra on three strands. The construction of grt ell shows that it is a Lie subalgebra of oder 2 , and that there is a canonical surjection (23) s : grt ell → grt.

Let r ell denote the kernel. Enriquez [E] showed that there also exists a Lie algebra morphism (24) γ : grt → grt ell that is a section of (23), i.e. such that γ • s = id on grt. Thus, there is a semi-direct product isomorphism (25) grt ell r ell γ(grt).

An elliptic version ds ell of the double shuffle Lie algebra ds was constructed in [S3] using mould theory, and it is shown there that like grt ell , ds ell is a Lie subalgebra of oder 2 , and that there is an injective Lie morphism γ : ds → ds ell that makes the diagram

grt → ds γ ↓ ↓ γ grt ell ds ell \ / oder 2 commute.
Our second main result on krv ell is an analog of the existence of γ and γ.

Theorem 7. There is an injective Lie algebra morphism γ : krv → krv ell .

Based on the known injective Lie morphisms grt → ds → krv evoked in §1.3 above, we believe that there are corresponding injective Lie morphisms between the elliptic versions of these Lie algebras. However, we were not able to prove that grt ell as defined in [E] injects into ds ell or krv ell . To circumvent this difficulty, we define a Lie subalgebra grt ell ⊂ grt ell , conjecturally isomorphic to grt ell , as follows.

Definition 7. For n ≥ 0, let δ 2n ∈ oder 2 denote the derivation of lie 2 defined by δ 2n (x) = ad(x) 2n (y), δ 2n ([x, y]) = 0.

Let b be the Lie subalgebra of oder 2 generated by the δ 2n .

Enriquez showed in [E] that δ 2n ∈ r ell for n ≥ 0, so b is a Lie subalgebra of r ell . Let B denote the normalization of b ⊂ r ell under the semi-direct action of γ(grt) on r ell of (25). We set (26) grt ell = B γ(grt).

Our third main result on krv ell relates all these maps via a commutative diagram.

Theorem 8. We have the following commutative diagram of injective Lie morphisms:

grt → ds → krv ↓ ↓ ↓ grt ell → ds ell → krv ell \ ↓ / oder 2 .
1.6. Outline of the article. In §2, we reformulate the defining conditions of krv, which lead to the first definition of lkv and the proof of Proposition 2. The next section, §3, gives a brief introduction to mould theory and a translation of the defining conditions of lkv into that language, and uses mould theory to prove Theorem 3. Finally, the proofs of Theorems 5, Theorem 7 and Theorem 8 are given in the three subsections of §4.

Reformulation of the definition of krv and definition of the linearized Lie algebra lkv

In this section, we give a convenient reformulation of the defining conditions of krv, which leads to a simple definition of the linearized version lkv that passes easily into the language of moulds which will be essential for our subsequent proofs in § §3,4.

2.1. The first defining condition of krv: specialness. The first of the two defining conditions of krv is that krv lies in sder 2 , i.e. elements of krv are special tangential derivations having the form

E a,b with E a,b (x) = [x, a], E a,b (y) = [y, b] and [x, a] + [y, b] = 0.
The following equivalent formulations of the property of specialness as properties of the polynomial b were given in [S1]. 2.2. The second defining condition of krv: divergence. We now consider the second defining condition of krv, the divergence condition. Because krv is weightgraded, we may restrict attention to derivations E a,b of homogeneous weight n, i.e. such that a and b are Lie polynomials of homogeneous degree n ≥ 3. The second defining condition (9) then simplifies to the existence of a constant c such that

tr(xa x + yb y ) = c tr (x + y) n -x n -y n in tr 2 .
Let us reformulate this as a condition only on b, just as we did for the first defining condition. Since a ∈ lie 2 , its trace is zero and thus tr(xa x ) = tr(a x x) = -tr(a y y) = -tr(ya y ), so tr(xa x + yb y ) = tr(yb y -ya y ). From Proposition 9, we have b y = b y , so now, using the circularity of the trace, the divergence condition can be reformulated as

tr (b y -b x )y = c tr (x + y) n -x n + y n .
We use this to express it as a condition directly on b y -b x as follows, using the push-operator defined in (12). Definition 8. A polynomial b ∈ Ass 2 of homogeneous weight n > 1 is said to be push-constant for the value c if b does not contain the monomial y n and for each 1 < r < n, writing b r for the depth r part of b, we have

r i=0 push i (b r ) = c w w
where the sum in the right-hand factor is over all monomials of weight n and depth r. Equivalently, b is push-constant if it does not contain y n and for all monomials w = x n , we have

v∈P ush(w) (b|v) = c
where (b|v) denotes the coefficient of the monomial v in b, and P ush(w) is the list (with possible repetitions) [w, push(w), . . . , push r (w)]. If c = 0, then b is said to be push-neutral. If b is a scalar multiple of x n , then b is push-neutral by default.

Example.

The simplest example of a push-constant polynomial is the sum of all monomials of a given depth, for example

b = x a yx b yx c + x c yx a yx b + x b yx c yx a + x a yx c yx b + x b yx a yx c + x c yx b yx a .
More interesting push-constant polynomials can be obtained from elements ψ ∈ grt by taking the projection of ψ onto the words ending in y and writing this as by. In this way we obtain for example:

b = 2x 2 y 2 - 11 2 xyxy + 9 2 xy 2 x - 1 2 yx 2 y + 2yxyx - 1 2 y 2 x 2 .
The following proposition shows that the divergence condition comes down to requiring that b y -b x be push-constant.

x )y) = c tr (x + y) n -x n -y n if and only if b y -b x is push-constant for the value nc. Furthermore, if this is the case then (27) c = 1 n (b | x n-1 y).
Proof. Let w be a monomial of degree n and depth r ≥ 1, and let C w denote the list of words obtained from w by cyclically permuting the letters, so that C w contains exactly n words (with possible repetitions). Let C y w denote the list obtained from C w by removing all words ending in x, so that C y w contains exactly r words. Write C y w = [u 1 y, . . . , u r y]. Then we have the equality of lists [u 1 , ..., u r ] = P ush(u 1 ).

Let c w = tr(w), i.e. c w is the equivalence class of w, which is the set of the words in the list C w , without repetitions: thus C w is nothing other than n/|c w | copies of c w . The divergence condition tr (b y -b x )y = c tr (x + y) n -x n -y n translates as the following family of conditions for one word in each equivalence class c w :

(28) v∈cw (b y -b x )y | v = c|c w |,
where each side is the coefficient of the class c w in the trace, i.e. the sum of the coefficients of the words in c w in the original polynomial.

If r > 1, we can choose a word uy ∈ C w that starts in y. Then from ( 121), the divergence condition on b implies that

c = 1 |c w | v∈cw (b y -b x )y | v = 1 n v∈Cw (b y -b x )y | v = 1 n v∈C y w (b y -b x )y | v = 1 n u ∈P ush(u) ((b y -b x ) | u ).
This is exactly the definition of b y -b x being push-constant for the value nc.

If r = 1, then w is of depth 1, |c w | = n and x n-1 y is the only word in c w ending in y. Thus (121) comes down to

(b y -b x )y | x n-1 y = nc.
But since b is a Lie polynomial, we have (b|x n ) = (b x |x n-1 ) = 0, so using b y = b y (by Proposition 9), we also have

(b y -b x )y | x n-1 y = (b y -b x | x n-1 ) = (b y |x n-1 ) = (b y |x n-1 ) = (b y y|x n-1 y) = (b|x n-1 y), which proves that nc = (b|x n-1 y) as desired. Note that this condition means that if b has no depth 1 part, then b y -b x is push-neutral.
We now have a new way of expressing krv, which is much easier to translate into the mould language. Definition 9. Let V krv be the vector space spanned by polynomials b ∈ lie C of homogeneous degree n ≥ 3 such that

(i) b is push-invariant, and (ii) b y -b x is push-constant for the value (b | x n-1 y) , equipped with the Lie bracket {b, b } = [b, b ] + E a,b (b ) -E a ,b (b)
where a and a are the (unique) partners of b and b respectively. Indeed, since Propositions 9 and 10 show that

krv ∼ → V krv E a,b → b (29)
is an isomorphism of vector spaces and krv is known to be a Lie subalgebra of sder 2 , the bracket on V krv is inherited directly from this and makes V krv into a Lie algebra.

2.3. The linearized Kashiwara-Vergne Lie algebra lkv. Using the above isomorphism of krv with the vector space V krv given by E a,b → b, let us now consider the depth-graded versions of the defining conditions of V krv , i.e. determine what these conditions say about the lowest-depth parts of elements b ∈ V krv . The pushinvariance is a depth-graded condition, so it restricts to the statement that the lowest depth part of b is still push-invariant; in particular, by Proposition 9 it admits of a unique partner a ∈ lie C such that [x, a] + [y, b] = 0, i.e. such that the associated derivation E a,b lies in sder 2 .

In the second condition, if b is of degree n and depth r = 1 and b 1 denotes the lowest-depth part of b, then (b 1 ) y = x n-1 , so the push-constance condition on b 1 is empty since (b 1 ) y = (b|x n-1 y)x n-1 . If r > 1, however, then (b|x n-1 y) = 0 and so the push-constance condition on b y -b x is actually push-neutrality, which implies the push-neutrality of (b r ) y alone, since (b r ) y is the only part of the expression b y -b x of minimal depth r -1. This leads to the following definition for the depth-graded version of the Kashiwara-Vergne Lie algebra.

Definition 10. The bigraded linearized Kashiwara-Vergne Lie algebra is defined by

lkv = b ∈ lie 2 (i) b is push-invariant (ii) b is circ-neutral if b is of depth > 1, ,
equipped with the bracket coming from the bracket of derivations in sder 2 , namely

{b, b } = [b, b ] + E a,b (b ) -E a ,b (b).
The proof of Proposition 1 above, that lkv is closed under the proposed Lie bracket, is deferred to the end of §4.1. Proposition 2, however, is proven by the very fact that the defining properties of lkv are properties held by the lowest-depth parts of elements of krv, since this means precisely that there is an injective linear map gr krv → lkv, which is a Lie morphism as both spaces are equipped with the Lie bracket coming from sder 2 . It is, however, an open question as to whether these two spaces are equal, since it is not clear that an element satisfying the defining conditions of lkv necessarily lifts to an element of krv. No examples of this are known, and it would be interesting to try to prove equality by starting with a polynomial lkv of depth r > 1 and finding a way to construct a depth by depth lifting to an element of krv.

Rational and polynomial moulds

In this section, we introduce the language of moulds and reformulate the defining conditions of lkv in this language. We end the section with the proof of Theorem 3 and its corollary in terms of moulds. We hope that this section and the next one, which explores the elliptic version of krv, will illustrate the way in which moulds are powerful tools in this context.

3.1. Moulds and alternality. For the purposes of this article, we are concerned only with rational function-valued moulds defined over the rationals. Écalle defines moulds with more general arguments and more general values, but in this article we will use the term mould merely to denote a collection A = A r (u 1 , . . . , u r ) r≥0 where each A r (u 1 , u 2 , ..., u r ) ∈ Q(u 1 , . . . , u r ), i.e. each A r is a rational function in r commutative variables u 1 , . . . , u r with coefficients in Q. The rational function A r is the depth r part of the mould. When the context is clear we sometimes drop the index and write A(u 1 , . . . , u r ) instead of A r (u 1 , . . . , u r ) for the depth r part. In particular we have

A 0 = A(∅) ∈ Q.
Moulds are equipped with addition and multiplication by scalars componentwise; thus they form a vector space. We write ARI for the subspace of (rational) moulds A with A(∅) = 0 (keeping in mind that this ARI is only a very small subspace of the full space of moulds studied by Écalle). For convenience, we also define the vector space ARI of moulds defined exactly like ARI except on a set of commutative

variables v 1 , v 2 , . . ., i.e. B ∈ ARI means B = B r r≥0 with B r ∈ Q[v 1 , . . . , v r ].
We say that a mould A is concentrated in depth r if A s = 0 for all s = r, and we let ARI r ⊂ ARI be the subspace of moulds concentrated in depth r. Thus ARI = ⊕ r≥1 ARI r .

We now introduce Écalle's important swap operator on moulds.

Definition 11. The swap operator maps ARI to ARI, and is defined by

swap B(v 1 , . . . , v r ) = B(v r , v r-1 -v r , . . . , v 1 -v 2 )
for B ∈ ARI. The inverse operator mapping ARI to ARI (which we also denote by swap, as the context is clear according to whether swap is acting on a mould in ARI or one in ARI) is given by

swap C(u 1 , . . . , u r ) = C(u 1 + • • • + u r , u 1 + • • • + u r-1 , . . . , u 1 )
for C ∈ ARI. Thus it makes sense to write swap • swap = id.

We also need to consider an important symmetry on moulds, based on the shuffle operator on tuples of commutative variables, which is defined by

Sh (u 1 , . . . , u i )(u i+1 , . . . , u r ) = (u σ -1 (1) , . . . , u σ -1 (r) ) | σ ∈ S i r ,
where

S i r is the subset of permutations σ ∈ S r such that σ(1) < • • • < σ(i) and σ(i + 1) < • • • σ(r). Definition 12. A mould A ∈ ARI is alternal if in each depth r ≥ 2 we have w∈Sh((u1,...,ui)(ui+1,...,ur)) A r (w) = 0 for 1 ≤ i ≤ r 2 .
By convention, the alternality condition is void in depth 1, i.e. all depth 1 moulds are considered to be alternal.

Example. In depth 4, there are two alternality conditions, given by

A(u 1 , u 2 , u 3 , u 4 ) + A(u 2 , u 1 , u 3 , u 4 ) + A(u 2 , u 3 , u 1 , u 4 ) + A(u 2 , u 3 , u 4 , u 1 ) = 0 A(u 1 , u 2 , u 3 , u 4 ) + A(u 3 , u 1 , u 2 , u 4 ) + A(u 3 , u 4 , u 1 , u 2 ) + A(u 1 , u 3 , u 2 , u 4 ) + A(u 1 , u 3 , u 4 , u 2 ) + A(u 3 , u 1 , u 4 , u 2 ) = 0
We write ARI al for the subspace of ARI consisting of alternal moulds.

3.2. Lie elements and alternal moulds. Alternality is important because alternal polynomial moulds correspond to Lie polynomials in the sense given in the following lemma, whose statements are well-known: the first one is a direct consequence of Lazard elimination (cf. Bourbaki), and for complete elementary proofs of all the statements, see [SST] or [S2].

Lemma 11. (i) The free associative algebra Ass 2 on x, y can be decomposed as a direct sum

Q x, y = Qx ⊕ Q C , where Ass C = Q C = Q C 1 , C 2 , . . . is the free non-commutative polynomial algebra on variables C i = ad i-1 x (y) for i ≥ 1. (ii) Let Ass r C denote the subspace of Ass C spanned by monomials C a1 • • • C ar . For each r ≥ 1, the map ma : Ass r C → ARI r C a1 • • • C ar -→ u a1-1 1 • • • u ar-1 r is a vector space isomorphism.
(iii) For each r ≥ 1, the map ma restricts to a vector space isomorphism

ma : lie r C → ARI r al , where lie r C = lie C ∩ Q r C .
Examples. The mould ma(C 3 ) = ma ([x, [x, y]]) is the mould concentrated in depth 1 given by u [x, y], y]) is the mould concentrated in depth 2 given by u 1 1 u 0 2 -u 0 1 u 1 2 = u 1 -u 2 . Definition 13. Let β denote the backwards writing operator on words in x, y, meaning that β(m) is obtained from a word m by writing it from right to left. The operator β extends to polynomials by linearity.

2 1 . Similarly, ma(C 2 C 1 -C 1 C 2 ) = ma([
We write ARI pol for the vector subspace of polynomial-valued moulds in ARI. Let us give the translation of the restriction of the swap operator to polynomialvalued moulds directly in terms of elements of Ass 2 (cf. [R] or [S2]). Let f ∈ Ass r C , and write f = xf x + yf y . Set g = β(yf y ), where β is the backwards operator of Definition 13. Thus all the monomials of g end in y. If we write g explicitly as where

u 0 = -u 1 -u 2 -• • •-u r . A mould B ∈ ARI is push-invariant if push (B) = B (in all depths).
The following proposition shows that this definition is precisely the translation into mould terms of the property of push-invariance for a Lie polynomial given in Definition 5 above. Let f = yb, so that b = f y . Recalling that y = C 1 , the associated moulds are related by the formula

(32) ma(f )(u 1 , ..., u r ) = ma(C 1 b) = u 0 1 ma(b)(u 2 , ..., u r ) = ma(b)(u 2 , ..., u r ). Since b ∈ (lie C ) n , we have β(b) = (-1) n-1 b. Set g = β(yf y ) = β(yb) = (-1) n-1 by = (-1) n-1 a k a x a1 y...yx ar y.
By (31), we have

swap ma(f ) (v 1 , . . . , v r ) = (-1) n-1 a k a v a1 1 ...v ar r .
Looking at push(b)y = a k a x ar yx a1 y • • • x ar-1 y, we see that push(b)y is obtained from by by cyclically permuting the groups x ai y. Since b = push(b) if and only if k (a1,...,ar) = k (ar,a1,...,ar-1) for each a, this is equivalent to

(33) swap ma(f ) (v 1 , ..., v r ) = swap ma(f ) (v r , v 1 , . . . , v r-1 ).
Using the definition of the swap, we rewrite (33) in terms of ma(f ) as

(34) ma(f )(v r , v r-1 -v r , . . . , v 1 -v 2 ) = ma(f )(v r-1 , v r-2 -v r-1 , . . . , v r -v 1 )
We now make the change of variables v r = u 1 + ...

+ u r , v r -v 1 = u r , v 1 -v 2 = u r-1 , . . . , v r-2 -v r-1 = u 2 , v r-1 = u 1 in this equation, obtaining (35) ma(f )(u 1 + • • • + u r , -u 2 -• • • -u r , u 2 , . . . , u r-1 ) = ma(f )(u 1 , u 2 , . . . , u r ).
Finally, using relation (32), we write this in terms of ma(b) as

(36) ma(b)(-u 2 -• • • -u r , u 2 , . . . , u r-1 ) = ma(b)(u 2 , . . . , u r ).
Making the variable change u i → u i-1 changes this to

(37) ma(b)(-u 1 -• • • -u r-1 , u 1 , . . . , u r-2 ) = ma(b)(u 1 , . . . , u r-1 ),
which is just the condition of mould push-invariance ma(b) in depth r -1.

3.4. Circ-neutrality and the second defining relation of lkv. Let us now show how to reformulate the second defining property of elements of lkv in terms of moulds.

Definition 14. Let circ be the mould operator defined on moulds in ARI by

circ(B)(v 1 , . . . , v r ) = B(v 2 , . . . , v r , v 1 ).
A mould B ∈ ARI is said to be circ-neutral if for r > 1 we have

r i=0 circ i (B)(v 1 , . . . , v r ) = 0.
If B is a polynomial-valued mould of homogeneous degree n (i.e. the polynomial

B(v 1 , . . . , v r ) is of homogeneous degree n -r for 1 ≤ r ≤ n), we say that B is circ-constant if r i=0 circ i (B)(v 1 , . . . , v r ) = c a 1 +•••+ar =d a i ≥0 v a1 1 • • • v ar r for all 1 < r ≤ n, where B(v 1 ) = cv n-1 1 . (If c = 0, then a circ-constant mould is circ- neutral.
) Correspondingly, we also say that a polynomial b ∈ Ass C of homogeneous degree n is circ-constant if, setting c = (b|x n-1 y), we have b = b 0 + c n y n where b y 0 is push-constant for the value c (cf. Definition 8). A polynomial-valued mould (resp. a polynomial in Ass C ) is said to be circ-constant if it is a sum of circ-constant homogeneous moulds (resp. polynomials).

Example. Let ψ ∈ grt be homogeneous of degree n. Then as we saw in the example following Definition 8, the polynomial ψ y is push-constant, so ψ y y is circ-constant. For example if n = 5, then ψ y y is given by

ψ y y = x 4 y -2x 3 y 2 + 11 2 x 2 yxy - 9 2 xyx 2 y + 3yx 3 y + 2x 2 y 3 - 11 2 xyxy 2 + 9 2 xy 2 xy - 1 2 yx 2 y 2 + 2yxyxy - 1 2 y 2 x 2 y -xy 4 + 4yxy 3 -6y 2 xy 2 + 4y 3 xy
which is easily seen to be circ-constant.

For an example of a circ-constant mould, we take B = swap ma(ψ) , which has the same coefficients as ψ y y: it is given by

B(v 1 ) = v 4 1 B(v 1 , v 2 ) = -2v 3 1 + 11 2 v 2 1 v 2 - 9 2 v 1 v 2 2 + 3v 3 2 B(v 1 , v 2 , v 3 ) = 2v 2 1 - 11 2 v 1 v 2 - 1 2 v 2 2 + 9 2 v 1 v 3 + 2v 2 v 3 - 1 2 v 2 3 B(v 1 , v 2 , v 3 , v 4 ) = -v 1 + 4v 2 -6v 3 + 4v 4 .
The following result proves that the circ-constance of a polynomial b and that of the associated mould ma(b) are always connected as in the example above. By additivity, it suffices to prove the result for b a homogeneous polynomial of degree n, so that the circ-constance of b is relative to just one constant c n = c = (b|x n-1 y).

k a v a1 1 • • • v ar r .
Observe that a polynomial is push-constant if and only it is also push-constant written backwards, so in particular, b y is push-constant if and only if β(b y ) is. Suppose that b is circ-constant, i.e. that b y and thus β(b y ) are push-constant for the value c = (b|x n-1 y). In view of (38), this means that a k a = c when a runs through the cyclic permutations of a = (a 1 , . . . , a r ) for every tuple a, and this in turns means precisely that the mould swap ma(b) is circ-constant. As for the circ-neutrality equivalence, it follows from the circ-constance, since circ-neutrality is nothing but circ-constance for the constant 0.

The notion of circ-constance will play a role later in §4.2. In this section we only need circ-neutrality. Indeed, we showed that a polynomial b lies in lkv, i.e. b is a Lie polynomial that is push-invariant and circ-neutral, if and only if the associated mould ma(b) is alternal (by Lemma 11 (iii)), push-invariant (by Proposition 12) and its swap is circ-neutral (by Proposition 13). In other words, we have shown that ma gives a vector space isomorphism (40) ma : lkv ∼ → ARI pol al+push/circneut , where the right-hand space is the subspace of ARI of polynomial-valued moulds in ARI that are alternal and push-neutral with circ-neutral swap. In fact this map is an isomorphism (41) lkv r n ARI r n-r ∩ ARI pol al+push/circneut , of each bigraded piece, where in general we write ARI r d for the subspace of polynomialvalued moulds of homogeneous degree d concentrated in depth r.

We will show at the end of §4.1 below that ARI pol al+push/circneut is a Lie algebra under the ari-bracket, and thus by the compatibility (104) of the ari-bracket with the Poisson bracket given below, we will then be able to conclude that lkv is also a Lie algebra, proving Proposition 1 of this paper. For all n ≥ 3 and r = 1, 2, 3, the map is an isomorphism of the bigraded parts ls r n lkv r n .

In order to prove this theorem, we first reformulate the statement in terms of moulds and give its proof. Let ARI al/al denote the space of moulds that are alternal and have alternal swap, and following Écalle's notation, let ARI al/al denote the subspace of ARI al/al of moulds that are even in depth 1. Directly from the definition of ls, we see that the map ma gives an isomorphism ma : ls

∼ → ARI pol al/al
onto the space of polynomial-valued moulds in ARI al/al . Therefore, Theorem 3 can be stated very simply in terms of moulds as ARI pol al/al ⊂ ARI pol al+push/circneut . We will actually prove the more general result without the polynomial hypothesis.

Theorem 14. There is an inclusion of mould subspaces

ARI al/al ⊂ ARI al+push/circneut , Moreover in depths r ≤ 3, we have ARI r ∩ ARI al/al = ARI r ∩ ARI al+push/circneut .
Proof. It is well-known that every alternal mould satisfies A(u 1 , . . . , u r ) = (-1) r-1 A(u r , . . . , u 1 ) (cf. [S2], Lemma 2.5.3) and that a mould that is al/al and even in depth 1 is also push-invariant (cf. [S2], Lemma 2.5.5). Thus in particular ARI al/al ⊂ ARI al+push . It remains only to show that a mould in ARI al/al is necessarily circ-neutral. In fact, since the circ-neutrality condition is void in depth 1, we will show that even a mould in ARI al/al is circ-neutral; the condition of evenness in depth 1 is there to ensure the push-invariance, but not needed for the circ-neutrality.

The first alternality relation is given by

A(u 1 , . . . , u r ) + A(u 2 , u 1 , . . . , u r ) + • • • + A(u 2 , . . . , u r , u 1 ) = 0.
Since A is push-invariant, this is equal to

push r A(u 1 , . . . , u r ) + push r-1 A(u 2 , u 1 , . . . , u r ) + • • • + push A(u 2 , . . . , u r , u 1 ) = 0.
But explicitly considering the action of the push operator on each term shows that

push i A(u 2 , . . . , u r-i , u 1 , u r-i+1 , . . . , u r ) = A(u i+1 , . . . , u r , u 0 , u 2 , . . . , u i ) = circ r-i A(u 0 , u 2 , . . . , u r ),
where

u 0 = -u 1 -• • • -u r , so this sum is equal to r-1 i=0 circ i A(u 0 , u 2 , . . . , u r ) = 0,
which proves that A is circ-neutral. This gives the inclusion.

Let us now prove the isomorphism in the cases r = 1, 2, 3. The case r = 1 is trivial since the alternality conditions are void in depth 1. A polynomial-valued mould concentrated in depth 1 is a scalar multiple of u d 1 , which is automatically in ARI al/al , and lies in ARI al/al if and only if d is even. Such a mould is automatically alternal and the circ-neutral condition is void; it is push-invariant thanks to the evenness of d. This shows that in depth 1, both spaces are generated by moulds u d 1 for even d, and are thus isomorphic. Now consider the case r = 2. Let A ∈ ARI pol al+push/circneut be concentrated in depth 2. The circ-neutral property of the swap is explicitly given in depth 2 by swap(A)(v 1 , v 2 ) + swap(A)(v 2 , v 1 ) = 0. But this is also the alternality condition on swap(A), so A ∈ ARI al/al . The isomorphism in depth 2 is thus trivial.

Finally, we consider the case r = 3. Let A ∈ ARI pol al+push/circneut be concentrated in depth 3, and let B = swap(A). Again, we only need to show that B is alternal, which in depth 3 means that B must satisfy the single equation ( 42)

B(v 1 , v 2 , v 3 ) + B(v 2 , v 1 , v 3 ) + B(v 2 , v 3 , v 1 ) = 0.
The circ-neutrality condition on B is given by ( 43)

B(v 1 , v 2 , v 3 ) + B(v 3 , v 1 , v 2 ) + B(v 2 , v 3 , v 1 ) = 0.
It is enough to show that B satisfies the equality

(44) B(v 1 , v 2 , v 3 ) = B(v 3 , v 2 , v 1 ),
since applying this to the middle term of (43) immediately yields the alternality property (42) in depth 3. So let us show how to prove (44).

We rewrite the push-invariance condition in the v i , which gives

B(v 1 , v 2 , v 3 ) = B(v 2 -v 1 , v 3 -v 1 , -v 1 ) (45) = B(v 3 -v 2 , -v 2 , v 1 -v 2 ) (46) = B(-v 3 , v 1 -v 3 , v 2 -v 3 ). ( 47
)
Making the variable change exchanging v 1 and v 3 , this gives

B(v 3 , v 2 , v 1 ) = B(v 2 -v 3 , v 1 -v 3 , -v 3 ) (48) = B(v 1 -v 2 , -v 2 , v 3 -v 2 ) (49) = B(-v 1 , v 3 -v 1 , v 2 -v 1 ). (50) By (45), the term B(v 2 -v 1 , v 3 -v 1 , -v 1 ) is circ-neutral with respect to the cyclic permutation of v 1 , v 2 , v 3 , so we have (51) B(v 2 -v 1 , v 3 -v 1 , -v 1 ) = -B(v 3 -v 2 , v 1 -v 2 , -v 2 ) -B(v 1 -v 3 , v 2 -v 3 , -v 3 ).
But the circ-neutrality of B also lets us cyclically permute the three arguments of B, so we also have

-B(v 3 -v 2 , v 1 -v 2 , -v 2 ) = B(-v 2 , v 3 -v 2 , v 1 -v 2 ) + B(v 1 -v 2 , -v 2 , v 3 -v 2 ).
Using (45) and substituting this into the right-hand side of (51) yields

B(v 1 , v 2 , v 3 ) = B(-v 2 , v 3 -v 2 , v 1 -v 2 ) + B(v 1 -v 2 , -v 2 , v 3 -v 2 ) -B(v 1 -v 3 , v 2 -v 3 , -v 3 ). (52)
Now, exchanging v 1 and v 2 in (50) gives

B(v 3 , v 1 , v 2 ) = B(-v 2 , v 3 -v 2 , v 1 -v 2 ),
and doing the same with (48) gives

B(v 3 , v 1 , v 2 ) = B(v 1 -v 3 , v 2 -v 3 , -v 3 ).
By identifying lie push C with the space of derivations that annihilate [x, y], this lemma shows that lie push C is a Lie algebra under the bracket of derivations. We state this as a corollary. Thus we know that lie push C is a Lie algebra and it contains the elliptic Kashiwara-Vergne space krv ell as a subspace. This leads to our first main result on krv ell .

Theorem 19. The subspace krv ell ⊂ lie push C is a Lie subalgebra.

In order to prove this theorem, we will make essential use of mould theory, and in particular, of the ari-bracket defined by Écalle that makes ARI into a Lie algebra ARI ari . The hairiest definitions and proofs have been relegated to Appendix 1, in order to streamline the exposition of the next paragraph, which contains some basic elements of mould theory that will lead to the proof of the theorem in 4.1.3.

4.1.2.

A few facts about moulds. In this paragraph we give a few brief reminders about some of the basic operators of mould theory and their connections with the familiar situation of lie 2 ; a very concise but self-contained exposition with full definitions is given in Appendix 1, and a complete exposition with proofs can be found in Chapters 2 and 3 of [S2]. In this section, we content ourselves with giving a list of mould operators that generalize the some of the most frequently considered operators on lie 2 such as the usual and the Poisson bracket, Ihara and special derivations, and the bracket , on lie push C . It is important to make the following two observations: (i) all these operators given in mould-theoretic terms can be applied to a much wider class of moulds than merely polynomial-valued moulds, which permits a number of proofs of results on polynomial-valued moulds (and thus polynomials in x, y) that are not accessible otherwise; (ii) there are some very important mould operators that are not translations of anything that can be phrased in the polynomial situation; this is where the real richness of mould theory comes into play. We do not use any of these in this section, but some of them will play a key role in the next subsection (see 4.2.4).

Recall from Lemma 11 that we have an injective linear map ma from Q C to polynomial-valued moulds which restricts to an isomorphism from lie C to alternal polynomial-valued moulds, i.e. • There is a Lie bracket lu on ARI satisfying ma [f, g] = lu ma(f ), ma(g) for f, g ∈ lie C . We write ARI lu for the Lie algebra ARI with this bracket.

• For each mould A ∈ ARI, there is a derivation arit(A) of ARI lu that corresponds to the Poisson or Ihara derivation on lie C in the sense that arit ma(f ) • ma(g) = -ma d f (g) . ari ma(f ), ma(g) = ma {f, g} .

We write ARI ari for the Lie algebra with this Lie bracket.

• There is a Lie bracket ari on ARI which satisfies the following relation with the ari-bracket in the special case where A and B are both push-invariant moulds:

(61) ari swap(A), swap(B) = swap ari(A, B) .

• There is a third Lie bracket on ARI, the Dari-bracket, which is obtained by transfer by the ∆-operator given in (56), i.e. it is given by ( 62)

Dari(A, B) = ∆ ari ∆ -1 (A), ∆ -1 (B) .
This means that ∆ gives an isomorphism of Lie algebras

(63) ∆ : ARI ari ∼ → ARI Dari .
• For each mould A ∈ ARI, there is an associated derivation Darit(A) of ARI lu that preserves ARI pol if A is polynomial-valued and satisfies the following property: the Dari-bracket of (62) can also be defined by Proof. The main point is the following result [BS] (see Theorem 3.5): if D 1 and D 2 lie in oder 2 , then the map

oder 2 → ARI ari (65) D → ∆ -1 ma D(x) , is an injective Lie morphism, i.e. ∆ -1 ma [D 1 , D 2 ](x) = ari ∆ -1 ma(D 1 (x)) , ∆ -1 ma(D 2 (x)) .
Applying ∆ to both sides of this and using (62), this is equivalent to We use Darit and Dari to prove the desired result in three steps as follows.

(66) ma [D 1 , D 2 ](x) = Dari ma D 1 (x) , ma D 2 (x) ,
Step 1. Since lie push C is the space of push-invariant Lie polynomials, we have ma(lie push C ) = ARI pol al+push . But we saw in Proposition 20 that lie push C is a Lie algebra under , , so ARI pol al+push is a Lie algebra under Dari.

Step 2. The space ARI ∆ al+push is a Lie algebra under ari. Indeed, the definition of ∆ shows that this operator does not change the properties of push-invariance or alternality, i.e. ∆ -1 (ARI al+push ) = ARI al+push . Restricted to polynomial-valued moulds, we have ∆ -1 (ARI pol al+push ) = ARI ∆ al+push . Since ∆ is an isomorphism from ARI ari to ARI Dari by virtue of ( 63) and ARI pol al+push is a Lie subalgebra of ARI Dari by Step 1, its image ARI ∆ al+push under ∆ -1 is thus a Lie subalgebra of ARI ari .

Step 3. We can now complete the proof of Theorem 19 by showing that the space ARI ∆ al+push * circneut is a Lie algebra under ari. For this, we need the following lemma, whose proof is deferred to the end of Appendix 1.

Lemma 21. The space ARI circneut of circ-neutral moulds A ∈ ARI forms a Lie algebra under the ari-bracket.

Given this, it is an easy matter to conclude. Let A, B lie in ARI ∆ al+push * circneut , and let us show that ari(A, B) lies in the same space. By Step 2, we know that ari(A, B) ∈ ARI ∆ al+push , so we only need to show that swap ari(A, B) is *circneutral. But we will show that in fact this mould is actually circ-neutral. To see this, let A 0 and B 0 be the constant-valued moulds such that swap(A) + A 0 and swap(B) + B 0 are circ-neutral. By Lemma 21, we have

ari swap(A) + A 0 , swap(B) + B 0 ∈ ARI circneut .
Using the identity swap ari(M, N ) = ari swap(M ), swap(N ) , valid whenever M and N are push-invariant moulds (cf. [S], (2.5.6)), as well as the fact that constantvalued moulds are both push and swap invariant, we have

ari swap(A) + A 0 , swap(B) + B 0 = ari swap(A + A 0 ), swap(B + B 0 ) = swap • ari(A + A 0 , B + B 0 ) = swap • ari(A, B) + swap • ari(A, B 0 ) + swap • ari(A 0 , B) + swap • ari(A 0 , B 0 ) = swap • ari(A, B)
since the definition of the ari-bracket shows that ari(C, M ) = 0 whenever C is a constant-valued mould. Thus swap • ari(A, B) is circ-neutral, which completes the proof of Theorem 19.

The following easy corollary provides the promised proof of Proposition 1 stating that lkv is a Lie algebra.

Corollary 22. The subspace

ARI pol al+push/circneut ⊂ ARI ∆ al+push * circneut
is a Lie algebra under the ari-bracket. Thus, by (60), the space

lkv = ma -1 ARI pol al+push/circneut
a Lie algebra under the Poisson bracket.

Proof. By the definition of ari, ARI pol is a Lie subalgebra of ARI. Also, Lemma 21 shows that the space ARI circneut of circ-neutral moulds is a Lie subalgebra of ARI * circneut . Thus ARI ∆ al+push/circneut is a Lie algebra inside ARI ∆ al+push * circneut . So the intersection

ARI pol ∩ ARI ∆ al+push/circneut = ARI pol al+push/circneut
is one as well.

4.2. The map from krv → 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 4.3 below is devoted to connections between these three situations.

Theorem 23. There is an injective Lie algebra morphism

(70) krv → krv ell
The proof constructs the morphism from krv to krv ell in 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 space V krv of Definition 9, via the map

ν : V krv ∼ → W krv (71) f → ν(f ), ( 72
)
where ν is the automorphism of Ass 2 defined by ( 73)

ν(x) = z = -x -y, ν(y) = y.
In paragraph 4.2.1, we prove that W krv is a Lie algebra under the Poisson or Ihara bracket, and give a description of W krv via two properties, the "twisted" versions of the two defining properties of V krv given in Definition 9.

Step 2. In paragraph 4.2.2, we study the mould space ma W krv . Thanks to the compatibility of the ari-bracket with the Poisson bracket ( 104), this space is a Lie subalgebra of ARI ari . Just as we reformulated the defining properties of lkv in mould terms in §3, proving that ma(lkv) = ARI pol al+push/circneut , here we reformulate the defining properties of W krv in mould terms: explicitly, we show that (74) ma W krv = ARI pol al+sen * circconst , the space of polynomial-valued moulds that are alternal, satisfy a certain senary relation (79) 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 degree n whose swap is circ-constant up to addition of a constant-valued mould, then the constant-valued mould B 0 is uniquely determined as being the mould concentrated in depth n and taking the value c/n there, where B(v 1 ) = cv n-1 1 .

Step 3. For this part we need to introduce Écalle's mould pal and its inverse invpal, which lie in the Lie group GARI associated to the Lie algebra ARI ari , and study the adjoint operator Ad ari (invpal) on ARI ari . Letting Ξ denote the map

Ad ari (invpal) • pari : ARI ari → ARI ari ,
we show that it yields an injective Lie morphism (75) Ξ : ARI pol al+sen * circconst → ARI ∆ al+push * circneut of subalgebras of ARI ari .

Step 4. The final step is to compose (75) with the Lie morphism ∆ : ARI ari → ARI Dari , obtaining an injective Lie morphism

ARI pol al+sen * circconst → ∆ ARI ∆ al+push * circneut ,
where the left-hand space is a subalgebra of ARI ari and the right-hand one of ARI Dari . Since the right-hand space is equal to ma(krv ell ), the desired injective Lie morphism krv → krv ell is obtained by composing all the maps described above, as shown in the following diagram:

krv by (29) ↓ V krv by (71) ↓ ν W krv krv ell
by ( 74) ↓ ma ma -1 ↑ by ( 57)

ARI pol al+sen * circconst Ξ -→ ARI ∆ al+push * circneut ∆ -→ ∆ ARI ∆ al+push * circneut by (75) 4.2.1.
Step 1: The twisted space W krv .

Proposition 24. Let W krv = ν(V krv ). Then W krv is a Lie algebra under the Poisson bracket.

Proof. The key point is the following lemma on derivations. Let us first show that

d ν(b) is the conjugate of E 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 on x and y, 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 to d ν(b) . To show that d ν(b) lies in ider 2 , we check that d ν(b) (z) is a bracket of z with another element of lie 2 : 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 by ν maps an element of ider 2 to an element of sder 2 , which yields the isomorphism (76) as vector spaces. To see that it is also an isomorphism of Lie algebras, it suffices to note that conjugation by ν preserves the Lie bracket of derivations in der 2 , i.e.

ν • [D 1 , D 2 ] • ν = [ν • D 1 • ν, ν • D 2 • ν],
since ν is an involution. Since the Lie brackets on sder 2 and ider 2 are just restrictions to those subspaces of the Lie bracket on the space of all derivations, conjugation by ν carries one to the other.

We use the lemma to complete the proof of Proposition 24. Write

krv ν = {ν • E • ν | E ∈ krv} ⊂ ider 2 .
By restricting the isomorphism (76) to the subspace krv ⊂ sder 2 , 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 ( 29) mapping E a,b → b, and the right-hand vertical map sends an Ihara derivation d f to f . Equipping W krv with the Lie bracket inherited from krv ν makes this into a commutative diagram of Lie isomorphisms. But this bracket is nothing other than the Poisson bracket since krv ν ⊂ ider 2 .

We now give a characterization of W krv by two defining properties which are the twists by ν of those defining V krv . Recall that β is the the backwards operator given in Definition 13.

Proposition 26. The space W krv is the space spanned by polynomials b ∈ lie C , of homogeneous degree n ≥ 3, such that

(i) b y -b x is anti-palindromic, i.e. β(b y -b x ) = (-1) n-1 (b y -b x ), and (ii) b + c
n y n is circ-constant, where c = (b|x n-1 y).

Proof. Let f = ν(b), so that f ∈ V krv . Then the property that b y -b x is antipalindromic is precisely equivalent to the push-invariance of f (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 since

f ∈ V krv , f y -f x is push-constant for the value c = (f |x n-1 y) = (-1) n-1 (b|x n-1 y). 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 since f (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: if g ∈ Ass C is a polynomial of homogeneous degree n that is push-constant for (-1) n-1 c, then ν(g) is push-constant for c, since taking g = f y -f x then shows that ν(g) = b y is push-constant for c. The proof of this statement is straightforward using the substitution z = -x -y (but see the proof of Lemma 3.5 in [S1] for details). Since c = 0 if f ∈ V krv is of even degree n (Corollary 4), this proves (ii).

4.2.2.

Step 2: The mould version ma(W krv ). The space ma(W krv ) is closed under the ari-bracket by (60), since W krv is closed under the Poisson bracket.

Let b ∈ W krv and let B = ma(b). Then since b is a Lie polynomial, B is an alternal polynomial mould. Let us give the mould reformulations of properties (i) and (ii) of Proposition 26. The second property is easy since we already showed, in Proposition 13, that a polynomial b is circ-constant if and only if swap(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 operator mantar defined in (54), as well as the mould operator pari defined by ( 77) pari(B)(u 1 , . . . , u r ) = (-1) r B(u 1 , . . . , u r ).

The operator pari extends the operator y → -y on polynomials to all moulds, and mantar extends the operator f → (-1) n-1 β(f ). Above all, we need Écalle's mould operator teru, defined by taking the mould teru(B) to be equal to B in depths 0 and 1, and for depths r > 1, setting

(78) teru(B)(u 1 , . . . , u r ) = B(u 1 , . . . , u r ) + 1 u r B(u 1 , . . . , u r-2 , u r-1 + u r ) -B(u 1 , . . . , u r-2 , u r-1 ) . Lemma 27. Let b ∈ lie C .
Then the following are equivalent:

(1) b y -b x is anti-palindromic;

(2) B = ma(b) satisfies the senary relation

(79) teru • pari(B) = push • mantar • teru • pari(B).
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 B = ma( b). Write b = bx x + by y as usual. Then for each depth part ( bx + by ) r of the polynomial bx + by (1 ≤ r ≤ n -1), the anti-palindromic property (80) ( fx + fy ) r = (-1) n-1 β( fx + fy ) r translates directly to the following relation on B:

(81) teru( B)(u 1 , . . . , u r ) = push • mantar • teru( B)(u 1 , . . . , u r ).
Let us deduce the equivalence of ( 1) and (2) from that of (80) and ( 81). Let b be defined by b(x, y) = b(x, -y). This implies that b x = (-1) rb x , b y = (-1) r-1b y , and B = pari (B). Thus b y -b x is anti-palindromic if and only if by + bx is, i.e. if and only if (80) holds for b, which is the case if and only if (81) holds for B, which is equivalent to (79) with B = pari (B). This proves the lemma.

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

Proposition 28. Let ARI pol al+sen * circconst denote the space of alternal polynomialvalued moulds satisfying the senary relation (79) and having swap that is circconstant up to addition of a constant-valued mould. Then we have the isomorphism of Lie algebras ma : W krv ∼ -→ ARI pol al+sen * circconst ⊂ ARI ari .

4.2.3. Mould background: Exponential maps from ARI to GARI. The next stage of our proof, the construction of a Lie algebra morphism (82) ARI pol al+sen * circconst → ARI ∆ al+push * circneut , is the most difficult, and requires some further definitions from mould-theory. In order to keep it simple, we will make use of the following scheme.

Any vector space g equipped with a pre-Lie law p(f, g) is is also automatically equipped with • a Lie bracket [f, g] = p(f, g) -p(g, f ); • an exponential map exp p : g → G, where G = exp p (g) is the associated Lie group, and its inverse map log p ;

• the group law * on G which is given by

exp p (f ) * exp p (g) = exp p ch [,] (f, g) .
• an adjoint map of G on g defined for H ∈ G by letting h = log p (H) and setting

Ad [,] (H) • f = exp ad(h) • f = n≥0 1 n! ad(h) n • f, where ad(h) • f = [h, f ].
When g = ARI, we have seen that it can be equipped with various pre-Lie laws and Lie brackets. The underlying set of the associated Lie group will always be the set GARI of all moulds with constant term 1, just as ARI is the space of all moulds with constant term 0. (The same holds for ARI and GARI.)

Écalle has studied a large family of different pre-Lie laws on ARI and ARI, together with all their attendant structures as in the list above. The only ones we need here are the pre-Lie laws

preari(A, B) = arit(B) • A + mu(A, B) on ARI preari(A, B) = arit(B) • A + mu(A, B) on ARI,
where arit (resp. arit) are the derivations of ARI lu (resp. ARI lu ) defined in Appendix 1. We will not use these pre-Lie laws in and of themselves, but in the next paragraph we will be using their associated adjoint actions Ad ari and Ad ari .

We end this paragraph by defining, for any mould Q ∈ GARI, an automorphism ganit(Q) of the Lie algebra ARI lu 2 . Set v = (v 1 , . . . , v r ), and let W v denote the set of decompositions d v of v into chunks ( 83)

d v = a 1 b 1 • • • a s b s
for s ≥ 1, where with the possible exception of b s , the a i and b i are non-empty. Thus for instance, when r = 2 there are two decompositions in W v , namely a 1 = (v 1 , v 2 ) and a 1 b 1 = (v 1 )(v 2 ), and when r = 3 there are four decompositions, three for s = 1:

a 1 = (v 1 , v 2 , v 3 ), a 1 b 1 = (v 1 , v 2 )(v 3 ), a 1 b 1 = (v 1 )(v 2 , v
3 ), and one for s = 2: a 1 b 1 a 2 = (v 1 )(v 2 )(v 3 ). Écalle's explicit expression for ganit(Q) is given by ( 84)

ganit(Q) • T (v) = a1b1•••asbs∈Wv Q( b 1 ) • • • Q( b s ) T (a 1 • • • a s ),
where if b i is the chunk (v k , v k+1 , . . . , v k+l ), then we use the notation

(85) b i = (v k -v k-1 , v k+1 -v k-1 , . . . , v k+l -v k-1 ).
4.2.4. Mould background: The special mould pal and Écalle's fundamental identity.

We are now ready to introduce the fundamental identity of Écalle, which is the key to the construction of the desired map (82).

Definition 16. Let constants c r ∈ Q, r ≥ 1, be defined by setting f (x) = 1 -e -x and expanding f * (x) = r≥1 c r x r+1 , where f * (x) is the infinitesimal generator of f (x), defined by

f (x) = exp f * (x) d dx • x.
Let lopil be the mould in ARI ari defined by the simple expression ( 86)

lopil(v 1 , . . . , v r ) = c r v 1 + • • • + v r v 1 (v 1 -v 2 ) • • • (v r-1 -v r )v r
Set pil = exp ari (lopil) where exp ari denotes the exponential map associated to preari, and set pal = 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 mould lopal = log ari (pal) is alternal (cf. [START_REF] Écalle | Eupolars[END_REF], or [S2], Chap. 4.). Thus the moulds pil and pal are both exponentials of alternal moulds; this is called being symmetral. The inverses of pal (in GARI) and pil (in GARI) 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 and pil, given by (87)

Ad ari (pal) = exp ad ari (lopal) , Ad ari (pil) = exp ad ari (lopil) ,

where ad ari (P ) • Q = ari(P, Q). The inverses of these adjoint actions are given by (88) Ad ari (invpal) = exp ad ari (-lopal) , Ad ari (invpil) = exp ad ari (-lopil) .

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's fundamental identity relates the two adjoint actions of (87). Valid for all push-invariant moulds M , it is given by ( 89)

swap • Ad ari (pal) • M = ganit(pic) • Ad ari (pil) • swap(M ),
where pic ∈ GARI is defined by pic(v 1 , . . . , v r ) = 1/v 1 • • • 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. Let poc ∈ GARI be the mould defined by ( 90)

poc(v 1 , . . . , v r ) = 1 v 1 (v 1 -v 2 ) • • • (v r-1 -v r )
.

Then ganit(poc) and ganit(pic) are inverse automorphisms of ARI lu (see [B], Lemma 4.37). Thus, we can rewrite the above identity (89) as (93) Ξ : ARI pol al+sen * circconst -→ ARI ∆ al+push * circneut . Proof. We have already shown that both spaces are Lie subalgebras of ARI ari , the first in Proposition 28 and the second in 4.1.3. Furthermore, since pari and Ad ari (invpal) are both invertible and respect the ari-bracket, the proposed map is indeed an injective map of Lie subalgebras. Thus it remains only to show that the image of ARI pol al+sen * circconst under Ξ really lies in ARI ∆ al+push * circneut . We will show separately that if B ∈ ARI pol al+sen * circconst and A = Ξ(B), then (i) A is push-invariant, (ii) A is alternal, (iii) swap(A) is circ-neutral up to addition of a constant-valued mould, (iv) A ∈ ARI ∆ .

Proof of (i). Écalle proved that Ad ari (pal) transforms push-invariant moulds to moulds satisfying the senary relation (81) (see [Ec] (3.58); indeed this is how the senary relation arose). Since B satisfies (79), B = pari(B) satisfies (81), so Ad ari (invpal)( B) = Ξ(B) = A is push-invariant.

Proof of (ii). The subspace of alternal moulds ARI al is closed under ari (cf. [SS]), so exp ari (ARI al ) forms a subgroup of GARI gari , which we denote by GARI as gari (the superscript as stands for symmetral). The pal is known to be symmetral (cf. [START_REF] Écalle | Eupolars[END_REF], or in more detail [S2], Theorem 4.3.4). Thus, since GARI as gari is a group, the gari-inverse mould invpal is also symmetral. Therefore the adjoint action Ad ari (invpal) on ARI restricts to an adjoint action on the Lie subalgebra ARI al of alternal moulds. If B is alternal, then pari(B) is alternal, and so A = 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 (92) given in 4.2.4, with N = pari(B) (recall that (92) is valid whenever Ad ari (invpal) • N is push-invariant, which is the case for pari(B) thanks to (i) above). The key point is that the operators ganit(poc) and Ad ari (pil) on the left-hand side of (92) are better adapted to tracking the circ-neutrality and the denominators than the right-hand operator Ad ari (invpal) considered directly.

Proof of (iii). Let b ∈ W krv , and assume that b is of homogeneous degree n. Let B = ma(b). Then by Proposition 26 and Proposition 13, swap(B) is circ-constant, and even circ-neutral if n is even.

We need to show that swap • Ad ari (invpal) • pari(B) is *circ-neutral. To do this, we use (92) with N = 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 that pari commutes with swap). We prove that this mould is *circ-neutral in three steps. First we show that the operator ganit(poc) • pari changes a circconstant mould into one that is circ-neutral (Proposition 30). Secondly, we show that the operator Ad ari (invpil) preserves the property of circ-neutrality (Proposition 32). Finally, we show that if M is a mould that is not circ-constant but only *circ-constant, and if M 0 is the (unique) constant-valued mould such that M + M 0 is circ-constant, then

Ad ari (invpil) • ganit(poc) • pari(M ) + M 0 is circ-neutral. Using (92), this will show that swap • Ad ari (invpal) • M is *circ- neutral.
Proposition 30. Fix n ≥ 3, and let M ∈ ARI be a circ-constant polynomial-valued mould of homogeneous degree n. Then ganit(poc) (83). For any decomposition d v , we let its b-part be the unordered set {b 1 , . . . , b s }, its a-part the unordered set {a 1 , . . . , a s }, and we write l a for the number of letters in the a-part, i.e.

• pari(M ) is circ-neutral. Proof. Let c = M (v 1 ) | v n-1 1 , and let N = pari(M ), so that N (v 1 ) = -cv n-1 1 . Let v = (v 1 , . . . , v r ), and let W v be the set of decompositions d v of v into chunks d v = a 1 b 1 • • • a s b s as in
l a = |a 1 | + • • • + |a s |. Let W = i W σ i r (v) ,
where the σ i r (v) are the cyclic permutations of v = (v 1 , . . . , v r ), and let W b denote the subset of decompositions in W having identical b-part. The decompositions in W having identical b-part to a given decomposition d v ∈ W v are as follows: there is exactly one decomposition in W σ i-1 r (v) for each i such that v i is one of the letters in the a-part of v, which is obtained from d v by placing dividers between the same letters. For example, if r = 5 and

d v = a 1 b 1 a 2 b 2 = (v 1 , v 2 )(v 3 )(v 4 )(v 5 ) then the two other decompositions having the same b-part {(v 3 ), (v 5 )} are given by (v 2 )(v 3 )(v 4 )(v 5 )(v 1 ) and (v 4 )(v 5 )(v 1 , v 2 )(v 3 ). Thus if b denotes the b-part of a given decomposition d v of v = (v 1 , . . . , v r ), then W b contains
exactly l a decompositions, more precisely exactly one decomposition of each cyclic permutation (v i , . . . , v r , v 1 , . . . , v i-1 ) with v i in the a-part of d v .

Also, for each n ≥ 1, let W a n denote the set of monomials w of degree n -l a in the letters lying in the a-part of d v . For instance in the example above

d v = (v 1 , v 2 )(v 3 )(v 4 )(v 5 ), the a-part is {(v 1 , v 2 ), (v 4 )} and W a 5 consists of all monomials of degree 2 in the three letters v 1 , v 2 , v 4 , i.e. W a 5 = {v 2 1 , v 2 2 , v 2 4 , v 1 v 2 , v 1 v 4 , v 2 v 4 }. Note in particular that W a n = {1} when |a| = n and W a n = ∅ when r > n.
The definition of lopil in (86) shows that lopil is trivially circ-neutral. Thus, since M is circ-neutral, ad ari (lopil) • M = ari(lopil, M ) is also circ-neutral by Lemma 21, and successively so are all the terms ad ari (lopil) n (M ). Thus Ad ari (invpil) • M is circ-neutral.

Finally, we now assume that swap(B) is a *circ-neutral polynomial-valued mould in ARI of homogeneous degree n. Let B 0 be the (unique) constant-valued mould such that swap(B) + B 0 is circ-neutral. Then by Propositions 30 and 32, the mould

Ad ari (invpil) • ganit(poc) • pari(B + B 0 )
is circ-neutral. This mould breaks up as the sum

Ad ari (invpil) • ganit(poc) • pari(B) + Ad ari (invpil) • ganit(poc) • pari(B 0 ),
but the operator Ad 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 + B 0 ) = Ad ari (invpil) • ganit(poc) • pari(B) + B 0 , 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 (92), this time to track the denominators that appear in the right-hand side. By (92), if B is a polynomialvalued mould satisfying the senary relation, and if

A = Ξ(B) = Ad ari (invpal) • pari(B), then A lies in ARI ∆ if and only if (98) swap • Ad ari (invpil) • ganit(poc) • swap pari(B) ∈ ARI ∆ .
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 applying Ad ari (invpil). The first result is that the denominators introduced by applying ganit(poc) are at worst of the form (v

1 -v 2 ) • • • (v r-1 -v r ). Lemma 33. Let M ∈ ARI pol . Then swap • ganit(poc) • M ∈ ARI ∆ .
Proof. The explicit expression for ganit(Q) given in (84) shows that the only denominators that can occur in ganit(poc) • M come from the factors

(99) poc( b 1 ) • • • poc( b s ) for all decompositions d v = a 1 b 1 • • • a s b 2 of v = (v 1 , . . . , v r ) into chunks as in (83), and b i = (v k -v k-1 , v k+1 -v k-1 , . . . , v k+l -v k-1
) (for k > 1) as in (85). Since poc is defined as in (90), the only factors that can appear in (99) are (v l -v l-1 ) where v l is a letter in one of b i , and these factors appear in each term with multiplicity one. Since the sum ranges over all possible decompositions, the only letter of v that never belongs to any b i is v 1 ; all the other factors (

v i -v i-1 ) appear. Thus (v 1 -v 2 )(v 2 -v 3 ) • • • (v r-1 -v r
) is a common denominator for all the terms in the sum defining ganit(poc) • M . The swap of this common denominator is equal to u 2 • • • u r , so this term is a common denominator for swap • ganit(poc) • M , which proves the lemma.

Lemma 34. Let M, N ∈ ARI * circneut be two moulds such that swap(M ) and swap(N ) lie in ARI ∆ . Then swap ari(M, N ) also lies in ARI ∆ . Proof. In Proposition A.1 of the Appendix of [BS], it is shown that if M and N are alternal moulds in ARI such that swap(M ) and swap(N ) lie in ARI ∆ , then swap ari(M, N ) also lies in ARI ∆ . In fact, it is shown in Proposition A.2 of that appendix that alternal moulds M whose swap lies in ARI ∆ satisfy the following property: setting

M (v 1 , . . . , v r ) = v 1 (v 1 -v 2 ) • • • (v r-1 -v r )v r M (v 1 , . . . , v r ),
we have (100) M (0, v 2 , . . . , v r ) = M (v 2 , . . . , v r , 0).

In fact, the proof that swap ari(M, N ) lies in ARI ∆ does not use the full alternality of M and N , but only (100). Therefore, the same proof goes through when M and N are *circ-neutral moulds such that swap(M ) and swap(N ) lie in ARI ∆ , as long as we check that every *circ-neutral mould M such that swap(M ) ∈ ARI ∆ satisfies (100).

To check this, let M be such a mould; by additivity, we may assume that M is concentrated in a single depth r > 1. This means that there is a constant C M such that M (v 1 , . . . , v r ) + M (v 2 , . . . , v r , v 1 ) + • • • + M (v r , v 1 , . . . , v r-1 ) = C M , which we can also write as M (v 1 , . . . , v r )

v 1 (v 1 -v 2 ) • • • (v r-1 -v r )v r + M (v 2 , . . . , v r , v 1 ) v 2 (v 2 -v 3 ) • • • (v r-1 -v r )(v r -v 1 )v 1 + • • • + M (v r , v 1 , . . . , v r-1 ) v r (v r -v 1 ) • • • (v r-2 -v r-1 )v r-1 = C M
where the numerators are polynomials. If we multiply the entire equality by v 1 and set v 1 = 0, only the first two terms do not vanish, and they yield precisely the desired relation (100).

Corollary 35. If N ∈ ARI is a *circ-neutral mould such that swap(N ) ∈ ARI ∆ , then also

(101) swap • Ad ari (invpil) • N ∈ ARI ∆ .
Proof. The lemma shows that swap • ari(lopil, N ) ∈ ARI ∆ since the mould lopil is circ-neutral and swap • lopil ∈ ARI ∆ by (86). In fact, applying the lemma successively shows that swap • ad ari (lopil) n (N ) ∈ ARI ∆ for all n ≥ 1. Since Ad ari (invpil) • N is obtained by summing these terms by (97), we obtain (101).

Appendix 1: Some facts on moulds

In this appendix, we introduce some mould definitions used in some of our proofs, and give the proof of Lemma 21.

Let ARI be the vector space of moulds with constant term 0. There are three different Lie brackets that one can put on the space ARI. We begin by introducing the standard mould multiplication that Écalle denotes mu(A, B): mu(A, B)(u 1 , . . . , u r ) = In order to define Écalle's ari-bracket, we first introduce three derivations of ARI lu associated to a given mould A ∈ ARI. It is non-trivial to prove that these operators are actually derivations (cf. [S2], Prop. 2.2.1).

  b,a (b ) -D b ,a (b), ã = D b,a (a ) -D b ,a (a).

E

  a,b (x) = [x, a], E a,b (y) = [y, b] and E a,b (z) = [z, c].The Lie bracket is explicitly given by(3) [E a,b , E a ,b] ] = E ãa, a ] + E a,b (a ) -E a ,b (a), b = [b, b ] + E a,b (b ) -E a ,b (b).• Let sder 2 denote the Lie subalgebra of tder 2 of special tangential derivations, i.e. derivations such that E a,b (z) = [x, a] + [y, b] = 0. • Let ider 2 be the Lie subalgebra of tder 2 of Ihara derivations, which are those that annihilate x, i.e. those of the form d b = E 0,b . The derivation d b is defined by its values on x and y (5) d b (x) = 0, d b (y) = [y, b]. The Lie bracket on ider 2 is given by [d b , d b ] = d {b,b } , where {b, b } is the Poisson (or Ihara) bracket given by (6) {b, b } = [b, b ] + d b (b ) -d b (b),

  E ā, b ∈ sder 2 where ā, b are the lowest-depth parts (i.e. the parts of lowest y-degree) of elements a, b ∈ lie 2 such that E a,b ∈ krv. If b is of homogeneous y-degree r, then ā is of homogeneous y-degree r + 1. Example. The smallest element of krv is in weight 3 and is given by E a,b with a = [[x, y], y], b = [x, [x, y]]. Since ā = a and b = b, this is also equal to E ā, b ∈ gr krv. The next smallest element of krv is in weight 5, and the depth-graded part E ā, b is given by

  = D b,a (b ) -D b ,a (b)

Proposition 9 .

 9 [Schneps,[START_REF] Schneps | Double shuffle and Kashiwara-Vergne Lie algebras[END_REF]] Let n ≥ 3 and let b ∈ lie C ; write b = b x x + b y y = xb x + yb y . Then the following are equivalent: (i) There exists a unique element a ∈ lie C such that [x, a] + [y, b] = 0; (ii) b is push-invariant; (iii) b y = b y . Thanks to this proposition, we can now reformulate the first defining condition of krv as follows: the pair of polynomials a, b ∈ lie C satisfies [x, a] + [y, b] = 0 if and only if b is push-invariant and a is its partner.

  Since E a,b ∈ sder, we have [x, a] = [b, y]. Expanding this in terms of the decompositions of a and b, we obtain xa x x + xa y y -xa x x -ya y x = xb x y + yb y y -yb x x -yb y y, from which we deduce that a y = b x and a y = b x . Thus tr(yb y -ya y ) = tr(yb y -yb x ) = tr y(b y -b x ) .

  Proposition 10. ([S1]) Let b be a push-invariant Lie polynomial of homogeneous degree n. Then b satisfies the divergence condition tr((b y -b

  ,...,ar) k a x a1 y • • • yx ar y, then (as shown in[S2], (3.2.6)), swap ma(f ) is the mould concentrated in depth r given by(31) swap ma(f) (v 1 , . . . , v r ) = a k a v a1 1 ...v ar r .3.3. Push-invariance and the first defining relation of lkv. Let us define the push-operator on moulds in ARI by (push B)(u 1 , . . . , u r ) = B(u 0 , u 1 , ..., u r-1 )

Proposition 12 .

 12 Let b ∈ lie C . Then b is a push-invariant polynomial if and only if ma(b) is a push-invariant mould. Proof. If b = y, then ma(b) is concentrated in depth 1 with value ma(b)(u 1 ) = 1, so these are both clearly push-invariant. Now let b ∈ (lie C ) r-1 n with n ≥ r ≥ 2. We write b = a=(a1,...,ar) k a x a1 y • • • yx ar .

Proposition 13 .

 13 Let b ∈ Ass C be of homogeneous weight n ≥ 3. Then b is a circ-constant polynomial if and only if swap ma(b) is a circ-constant mould, and b is circ-neutral if and only if swap ma(b) is circ-neutral. Proof. Let β be the backwards-writing operator on Ass C (cf. Definition 13). Write b = xb x + yb y , and let g = β(yb y ) = β(b y )y. For r ≥ 1, let g r denote the depth r part of g. If we write the polynomial g r as (38) g r = β (b y ) r-1 y = a=(a1,...,ar) k a x a1 y • • • yx ar y, then we saw in (30) and (31) that (39) swap ma(b) (v 1 , . . . , v r ) = a=(a1,...,ar)

3. 5 .

 5 Proof of Theorem 3. Recall the statement of Theorem 3. Theorem 3. The Lie injection (14) extends to a bigraded Lie injection on the associated linearized spaces, giving the following commutative diagram: gr ds → gr krv. ↓ ↓ ls → lkv.

Corollary 18 .

 18 The map b → D b,a gives an isomorphism (58) ∂ : lie push C → oder 2 whose inverse is D b,a → D b,a (x) = b, and this becomes a Lie isomorphism when lie push C is equipped with the Lie bracket (59) b, b = [D b,a , D b ,a ](x) = D b,a (b ) -D b ,a (b).

  The precise definitions of all the Lie brackets and derivations below are given in Appendix 1.

•

  There is a Lie bracket ari on ARI given by ari(A, B) = lu(A, B) -arit(A) • B + arit(B) • A that corresponds to the Poisson or Ihara bracket on lie C in the sense that (60)

→

  B) = Darit(A) • B -Darit(B) • A. We end this section by comparing the Dari-bracket to the bracket , on lie push C given in Corollary 18. Proposition 20. The map ma : lie push C ARI Dari , is a Lie algebra morphism, i.e. the Lie brackets , and Dari are compatible in the sense that ma b, b = Dari ma(b), ma(b ) .

∼→→

  ma : oder 2 → ARI Dari is a Lie algebra morphism. We saw in Corollary 18 that we have a Lie isomorphism lie push C oder 2 when lie push C is equipped with the Lie bracket (59), so by composition, we have an injective Lie morphism b → D b,a Ψ → ∆ -1 ma(D b,a (x)) ∆ → ma(b) is an injective Lie morphism lie push C ARI Dari , which proves the result. 4.1.3. Proof that krv ell is a Lie algebra. This subsection is devoted to the proof of Theorem 19, i.e. that the subspace krv ell ⊂ lie push C is closed under the bracket , . From Proposition 20, ma gives an injective Lie algebra morphism lie push C → ARI Dari . Thus it is equivalent to prove that the image of the subspace krv ell ⊂ lie push C is closed under the Dari-bracket. Since we saw above that ∆ -1 : ARI Dari → ARI ari , it is equivalent to show that ARI ∆ al+push * circneut is a Lie subalgebra of ARI ari . Let b ∈ lie C be push-invariant and let D b,a = ∂(b) where ∂ : lie push C → oder 2 is as in (58). It is shown 1 in [BS], Prop. B.1 that for all b ∈ lie C , we have (68) ma D b,a (b ) = Darit ma(b) ma(b ) . Thus when b ∈ lie push C and B = ma(b), Darit(B) is nothing but the mould form of D b,a ; in particular Darit(B) preserves the space of polynomial-valued moulds, Darit(B) • ma([x, y]) = 0 and Darit(B) • ma(y) = ma(a). This shows in particular that if b, b ∈ lie C and B = ma(b), B = ma(b ), then by (64) and (68), we have Dari(B, B ) = Darit(B) • B -Darit(B ) • B = ma D b,a (b ) -D b ,a (b) = ma [D b,a , D b ,a ](x) = ma b, b .(69)

Lemma 25 .

 25 Conjugation by ν induces an isomorphism of Lie algebras b → d ν(b) . Proof. Recall that E a,b ∈ sder 2 maps x → [x, a] and y → [y, b], and d ν(b) ∈ ider 2 is the Ihara derivation defined by x → 0, y → [y, ν(b)] (cf. §1.1).

2

  The explicit expression given below does not explain why ganit(Q) is an automorphism. The mould-theoretic definition of ganit(Q) makes this clear. Let anit be the derivation of ARI given in Appendix 1. Then preani(A, B) = anit(B) • A -mu(A, B) is a pre-Lie law on ARI. Let log ani be the associated logarithm map, and set P = log ani (Q) ∈ ARI. Then ganit(Q) is the exponential of the derivation anit(P ).

  ) • swap • Ad ari (pal) • M = Ad ari (pil) • swap(M ), and letting N = Ad ari (pal) • M , i.e. M = Ad ari (invpal) • N , we rewrite it in terms of N as (92)Ad ari (invpil)• ganit(poc) • swap(N ) = swap • Ad ari (invpal) • N,this identity being valid whenever M = Ad ari (invpal) • N is push-invariant.4.2.5.Step 3: Construction of the map Ξ. In this section we finally arrive at the main step of the construction of our map krv → krv ell , namely the construction of the map Ξ given in the following proposition.Proposition 29. The operator Ξ = Ad ari (invpal) • pari gives an injective Lie morphism of Lie subalgebras of ARI ari :

  To conclude, we set M = swap • pari(B); then by Lemma 33 we haveswap • ganit(poc) • swap • pari(B) ∈ ARI ∆ .By Proposition 30 this mould is *circ-neutral, so we can apply Corollary 35 with N = ganit(poc) • swap • pari(B) to conclude thatswap • Ad ari (invpil) • ganit(poc) • swap • pari(B) ∈ ARI ∆ .Thus thus by (92) with N = pari(B), we finally find thatAd ari (invpal) • pari(B) = Ξ(B) ∈ ARI ∆ ,which completes the proof of (iv).We have thus finished proving Proposition 29. Backtracking, this means we have completed the details of Step 3 of the proof of Theorem 23. Step 4, the final step in the proof, is very easy and was explained completely just before paragraph 4.2.1. Thus we have now completed the proof of Theorem 23, i.e. we have completed the construction of the injective Lie algebra morphism krv → krv ell .4.3. Relations with elliptic Grothendieck-Teichmüller and double shuffle.The final result in this paper is the proof of Theorem 8. In fact, this result is simply a consequence of putting together the results of the previous sections with known results. Indeed, the commutativity of the diagramgrt → ds ↓ ↓ (102) grt ell → ds ell \ / oder 2 ,where Ad ari (invpal) : ds → ds ell is the right-hand vertical map is shown in[S3].By (10), the injective map ds → krv is given by b(x, y) → b = b(z, -y), or more precisely to the derivation in krv given by a → b, [a, b] → 0.If b(x, y) ∈ ds, then b(x, -y) lies in W krv and b(z, -y) lies in V krv , so this map unpacks to ds y →-y→ W krvx →z → V krv → krv, where the last map comes from (29). We can thus construct a commutative square∆ al * al ) ⊂ ∆(ARI ∆ al+push * circneut ) ma -1 ↓ ↓ ma -1ds ell ⊂ krv ell . The first line of this diagram comes from the injection ds → krv and the definition of W krv . The second line is the direct mould translation of the top one, as the left-hand space is exactly ma(ds), the right-hand space is ma(W krv ) by (74), and the map pari restricted to polynomials is nothing other than y → -y. The vertical morphism Ad ari (invpal) : ARI pol al * il → ARI ∆ al * al is proven in [S3], and the vertical morphism Ad ari (invpal) • pari : ARI pol al+sen * circconst → ARI ∆ al+push * circneut comes from Proposition 29. Since pari is an involution, this proves that the horizontal injection in the third line of the diagram is nothing but an inclusion. Finally, the last line of the diagram comes from the definitions ds ell = ∆(ARI ∆ al * al ) ([S3]) and krv ell = ∆(ARI ∆ al+push * circneut ) by Definition 15. This diagram shows that the diagram (102) above can be completed by the diagram (103) to the commutative diagram of Theorem 8.

  1 , . . . , u i )B(u i+1 , . . . , u r ).The associated Lie bracket lu is defined by lu(A, B) = mu(A, B) -mu(B, A). We write ARI lu for ARI viewed as a Lie algebra for the lu-bracket. The identical formulas yield a multiplication and Lie algebra (also called mu and lu) on ARI. If f and g are power series in Ass C and A = ma(f ), B = ma(g), then mu is a mould translation of the usual non-commutative multiplication, and lu the usual Lie bracket: mu(A, B) = ma(f g), lu(A, B) = ma [f, g] .

  Definition 17.[Ec] Let B ∈ ARI. Then the derivation amit(B) of ARI lu is given byamit(B)•A (u 1 , . . . , u r ) = 1≤i<j<r A(u 1 , . . . , u i , u i+1 +• • •+u j+1 , u j+2 , . . . , u r )B(u i+1 , . . . , u j ),and the derivation anit(B) is given byanit(B)•A (u 1 , . . . , u r ) = 1<i<j≤r A(u 1 , . . . , u i +• • •+u j , u j+1 , . . . , u r )B(u i+1 , . . . , u j ).We also have corresponding derivations amit(B) and anit(B) of ARI lu for B ∈ ARI, given by the formulasamit(B)•A (v 1 , . . . , v r ) = 1≤i<j<r A(u 1 , . . . , u i , u j+1 , . . . , u r )B(u i+1 -u j+1 , . . . , u j -u j+1 ), anit(B)•A (v 1 , . . . , v r ) = 1<i<j≤r A(u 1 , . . . , u i , u j+1 , . . . , u r )B(u i+1 -u i , . . . , u j -u i ).Finally, Écalle defines the derivation arit(B) on ARI lu by arit(B) = amit(B) -anit(B), and the ari-bracket on ARI by ari(A, B) = arit(B) • A -arit(A) • B + lu(A, B), as well as the derivation arit on ARI lu and the bracket ari on ARI by the same formulas with overlines. Remark. The definitions of amit, anit, arit and ari are generalizations to all moulds of familiar derivations of Ass C . Indeed, if b, b ∈ Ass C and A = ma(b), B = ma(b ), then amit(B) • A = ma D l g (f ) where D l g is defined by x → 0, y → b y, anit(B) • A = ma D r b (b) where D r b is defined by x → 0, y → yb , and thus arit(B) • A = ma -d b (b) where d g is the Ihara derivation x → 0, y → [y, b ] (see (5)), and (104) ari(A, B) = ma [b, b ] + d b (b ) -d b (b) = ma {b, b } . corresponds to the Ihara or Poisson Lie bracket (6) on lie C . (See [S2], Corollary 3.3.4).

  i-1 (y) for i ≥ 1, and let lie C denote the degree completion of the Lie algebra L[[C 1 , C 2 , . . .]] on the C i . By Lazard elimination, lie C is free on the C i and

	(11)

Note that the notation is slightly different there; we recover this statement by setting F = b , U = b, D U = D b,a and taking care to note that the definition of Darit U in that article is the conjugation of the definition (107) used here by dar, i.e. it is (107) without the dar terms.

Substituting these two expressions as well as (49) into the right-hand side of (52), we obtain the desired equality (44). This concludes the proof of Theorem 3.

Remark. We conjecture that the inclusion of Theorem 14 is an isomorphism. But even the proof of the simple equality ( 44) is surprisingly complicated in depth 3, let alone in higher depth. Computer calculation does lead to the general conjecture: Conjecture. If A ∈ ARI al+push/circneut and B = swap(A), then for all r > 1, we have (53) B(v 1 , . . . , v r ) = (-1) r-1 B(v r , . . . , v 1 ).

The identity (53) would also yield the following useful partial result, which is the mould analog for lkv of a result that is well-known for ls, namely that the bigraded part ls r n = 0 when n ≡ r mod 2. Lemma 15. Fix 1 ≤ r ≤ n. Let A ∈ ARI r n-r ∩ ARI pol al+push/circneut and let B = swap(A). Assume that B satisfies (53). Then if n -r is odd, A = 0.

Proof. Let mantar denote the operator on moulds in ARI (resp. ARI) defined by ( 54) mantar(A)(u 1 , . . . , u r ) = (-1) r-1 A(u r , . . . , u 1 ) (resp. the same expression with v i instead of u i ). It is easy to check the following identity of operators noted by Écalle:

where (55) neg(A)(u 1 , . . . , u r ) = A(-u 1 , . . . , -u r ).

Let A ∈ ARI al+push/circneut ; then A is push-invariant, so applying the left-hand operator to A gives neg(A). Assuming (53) for B = swap(A), i.e. assuming that B = mantar (B), we see that applying the right-hand operator to A fixes A since on the one hand swap • swap = id and on the other, mantar(A) = A for all alternal moulds (cf. [S2], Lemma 2.5.3). Thus A must satisfy neg(A) = A, i.e. if A = 0 then the degree d = n -r of A must be even.

This implies the following result, which is the analogy for lkv of the similar well-known result on ls.

Corollary 16. If the swaps of all elements of ARI pol al+push/circneut are mantarinvariant, then ARI r d ∩ ARI pol al+push/circneut = 0 whenever d is odd, i.e. by ( 41), lkv r n = 0 when n ≡ r mod 2 4. The elliptic Kashiwara-Vergne Lie algebra

In this section we follow the procedure of [S3] for the double shuffle Lie algebra to define a natural candidate for the elliptic Kashiwara-Vergne Lie algebra, closely related to the linearized Kashiwara-Vergne Lie algebra, and give some of its properties.

4.1. Definition of the elliptic Kashiwara-Vergne Lie algebra.

4.1.1. The Kashiwara-Vergne Lie algebra. Let ∆ be the mould operator given by ( 56)

for r ≥ 1, and let ARI ∆ denote the space of rational-function moulds A such that ∆(A) is a polynomial mould (i.e. the denominator of the rational function A is "at worst"

. We write ARI ∆ * for the space of moulds in ARI ∆ ∩ ARI * , where * may represent any (or no) properties on moulds in ARI.

Recall that earlier we used the notation ARI a/b for the space of moulds having property a and whose swaps have property b; for example, ARI al/al denotes the space of alternal moulds with alternal swap. In this section we introduce a slightly more general notation ARI a * b to denote the space of moulds having property a and whose swap has property b up to adding on a constant-valued mould; thus, we write ARI al * al for the space of alternal moulds whose swaps are alternal up to adding on a constant-valued mould. An example of a mould in ARI al * al is the mould ∆ -1 (A), where A is the polynomial mould concentrated in depth 3 given by

It is easy to check that ∆ -1 (A) is alternal, but its swap is not alternal unless one adds on the constant 1/3.

Definition 15. The mould elliptic Kashiwara-Vergne vector space is the subspace of polynomial-valued moulds ∆ ARI ∆ al+push * circneut . The elliptic Kashiwara-Vergne vector space is the subspace krv ell ⊂ lie C such that (57) ma krv ell = ∆ ARI ∆ al+push * circneut .

The operator ∆ trivially respects push-invariance of moulds, so the space krv ell lies in the space lie push C of push-invariant elements of lie C . We will now show that the subspace krv ell is actually a Lie subalgebra of lie push C , which is itself a Lie algebra thanks to the following lemma, of which a more explicit version (with a formula for the partner) is proved in [S3] We now consider the mould N = pari(M ), of fixed homogeneous degree n, with

By the explicit formula (84), we have

so adding up over the cyclic permutations of v, we have

where the last equality follows from (94).

If c = 0, the expression ( 96) is trivially equal to zero in all depths r > 1, so we obtained the desired result that ganit(poc) • pari(M ) is circ-neutral. In order to deal with the case where M is circ-constant for a value c = 0, we use a trick and subtract off a known mould that is also circ-constant for c.

Lemma 31. For n > 1 and any constant c, let T n c be the homogeneous polynomial mould of degree n defined by

where P r n is the sum is over all monomials of degree n -r in the variables v 1 , . . . , v r for 1 ≤ r ≤ n. Then T n c is circ-constant and ganit(poc) • pari(T n c ) is circ-neutral. The proof of this lemma is annoyingly technical, so we have relegated it to Appendix 2. Consider the mould N = M -T n c . The mould N is circ-constant since M and T n c both are, but N (v 1 ) = 0, so by the result above, we know that

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

Proof. By (88), we have

We now pass to the Dari-bracket, which is the Lie bracket on ARI obtained by transfer by the ∆-operator given in (56): it is given by ( 105)

This means that ∆ gives an isomorphism of Lie algebras

It is shown in [S3], Prop. 3.2.1 that we have a second definition for the Daribracket, which is more complicated but sometimes very useful in certain proofs. Let dar denote the mould operator defined by dar

We begin by introducing, for each A ∈ ARI, an associated derivation Darit(A) of ARI lu by the following formula:

where ad(A) B). Then Dari corresponds to the bracket of derivations, in the sense that ( 108)

We are now armed to attack Lemma 21, whose statement we recall.

Lemma 21. The space ARI circneut of circ-neutral moulds A ∈ ARI forms a Lie algebra under the ari-bracket.

Proof. Let A, B ∈ ARI circneut . We need to show that

where the formula for the ari-bracket on ARI is given as in 4.1.3 by the expression

We will show that this expression is circ-neutral because in fact, each of the five terms in the sum is individually circ-neutral. Let us start by showing this for the first term, lu(A, B).

Let σ denote the cyclic permutation of {1, . . . , r} defined by

By additivity, since the circ-neutrality property is depth-by-depth, we may assume that A is concentrated in depth s and B in depth t, with s ≤ t, s + t = r. In this simplifed situation, we have

We have

as the terms cancel out pairwise.

We now prove that the second term

is circ-neutral. Fix j ∈ {1, . . . , s} and consider the term

Thus for each of the other terms

in the sum, with i ∈ {1, . . . , s}, there is exactly one cyclic permutation, namely σ j-i , that maps this term to A(v σ j-i (1) , . . . , v σ j-i (i-1) , v σ j-i (i+t , . . . , v σ j-i (r) )B(v j -v j+t , . . . , v j+t-1 -v j+t ).

For fixed j ∈ {1, . . . , s}, the values of k = j-i mod s as i runs through {1, . . . , s} are exactly {0, . . . , s-1}. Therefore, the coefficient of the term B(v j -v j+t , . . . , v j+t-1v j+t ) in the sum of the cyclic permutations of amit(B) • A is equal to

which is zero due to the circ-neutrality of A. Thus the coefficient of the term B(v j -v j+t , . . . , v j+t-1 -v j+t ) in the sum of the cyclic permutations of amit(B) • A is zero, and this holds for 1 ≤ j ≤ s, so the entire sum is 0, i.e. amit(B) • A is circ-neutral. The proof of the circ-neutrality of the term anit(B) • A is analogous. By exchanging A and B, this also shows that amit(A) • B and anit(A) • B are circ-neutral, which concludes the proof of the lemma.

Appendix 2: Proof of Lemma 31

Let us recall the statement of the technical lemma 31.

Lemma 31. For n > 1 and any constant c = 0, let T n c be the homogeneous polynomial mould of degree n defined by

, where P r n is the sum is over all monomials of degree n-r in the variables v 1 , . . . , v r . Then T n c is circ-constant and ganit(poc

Proof. The mould T n c is trivially circ-constant. Consider the proof of Proposition 30 with M = T n c , N = pari(M ). In order to show that ganit(poc)•N is circ-neutral, we start by recalling from the proof of Proposition 30 that for each r > 1, the cyclic sum ganit(poc)

) is equal to the expression (96). Thus we need to show that ( 96) is equal to zero for all r > 1. To show this, we will break up the sum (109) b={b1,...,bs}

into parts that are simpler to express.

We need a little notation. Let us write V j = {1, . . . , j}. If B ⊂ V r , let P d B denote the sum of of all monomials of degree d in the variables v i ∈ B. We write P 0 B = 1 for all B.

We will break up the sum (109) as the sum of partial sums S 0 + • • • + S r , where S 0 is the term of (109) corresponding to the empty set and S i is the sum over the b-parts containing v i but not v i+1 , . . . , v r , for each i ∈ {1, . . . , r}. Notice that the b-parts containing v i but not v i+1 , . . . , v r are in bijection with the 2 i-1 subsets B ⊂ V i-1 , by taking b to be the set B = B ∪ {v i }, divided into chunks consisting of consecutive integers. For example, if i = 5 and

Setting v 0 = v r , this means that S 0 = P n-r Vr and for 1 ≤ i ≤ r, (110)

.

In order to prove that ( 109) is zero, we will give simplified expressions for S 1 , . . . , S r-1 in Claim 1, a simplified expression for S r in Claim 2, and then show how to sum them up in Claim 3.

Claim 1. For 1 ≤ i ≤ r -1, we have v i+1 , . . . , v r . Let v 0 = v r . Then we have ( 111)

Proof. We will use the following trivial but useful identity. Let B V r , let v j ∈ B, and let B = B ∪ {v j }. Then (112)

Multiplying by the common denominator, we write (110) as

We will show below that for each 1 ≤ k ≤ i -1, the right-hand side of ( 113) is equal to the expression

Taking k = i -1 in this expression, the sum Q i-1 reduces to the single term corresponding to B = ∅, which is just P n-r+i vi-1,vi+1,...,vr-1 . Thus by ( 113), we obtain

which proves (111).

Let us prove that the right-hand side of ( 113) is equal to Q k for all 1 ≤ k ≤ i -1. We will use induction on k. Let us do the base case k = 1 by showing that the right-hand side of ( 113) is equal to Q 1 . We start by breaking the right-hand side of (113) into v 1 ∈ B and v 1 ∈ B, and compute

, which is exactly Q 1 . The last equality is obtained by using (112) twice on the right-hand factor. This proves the base case k = 1. Now fix k < i -1 and assume that

We will show by the same method that Q k = Q k+1 . We break the expression for

again by (112) applied twice. Thus

and thus the right-hand side of ( 113) equals Q i-1 as desired.

Unfortunately, the expression in Claim 1 for S i does not work for i = r due to the fact that when i = r in (113), the subset B = V r-1 occurs in the sum and the corresponding polynomial P Vr\B = 0. It turns out that the expression for S r is actually simpler.

Claim 2. The term S r is given by ( 114)

) .

Proof. We will show that (115)

starting from the equality (113) for i = r, slightly rewritten as

.

We have P ∅ = 0, and we may as well sum over the subsets C, so it becomes

.

We will prove the following formula, valid for 1 ≤ i ≤ r -1 and n ≥ 1:

where we set P 0 B = 1 and P m B = 0 if m < 0.

This equality suffices to prove the desired result (115). Indeed, taking i = r -1, we see that R n r-1 is equal to the right-hand side of (118), so r j=1

and canceling out the factor (v r -v r-1 ) from both sides yields (115).

Let us prove (119) by induction on i. When i = 1, we have B = {v i } and for all n ≥ 1, we have

, proving the base case. Assume (119) holds for i -1 for all n ≥ 1. Fix n. We break R n i into the sum over B containing v i and B not containing V i , as follows:

The key point is that for B not containing v i , we can write

Using this, the equality becomes

This proves (119) and thus completes the proof of Claim 2.

We can now prove that the expression ( 109) is equal to zero by showing that S 0 + • • • + S r = 0. Claim 3. We have S 0 + • • • + S r = 0.

Proof. The key point is the following computation of partial sums for i < r:

(120) S 0 + • • • + S i = (-1) r-i P n-r+i vi,...,vr-1

We prove it by induction on i. The base case i = 0 is just given by the formula for S 0 (with v 0 = v r ). Assume (120) up to i -1. Then

(-1) r-i+1 P n-r+i-1 vi-1,...,vr-1

) + (-1) r-i P n-r+i vi-1,vi+1,...,vr-1

Using (112) and multiplying (121) by the denominator, we find

vi-1,vi+1,...,vr-1 -(v i-1 -v i )P n-r+i-1 vi-1,...,vr-1 = P n-r+1 vi-1,vi+1,...,vr-1 + v i P n-r+i-1 vi-1,...,vr-1 -v i-1 P n-r+i-1 vi-1,...,vr-1 = P n-r+1 vi-1,...,vr-1 -v i-1 P n-r+i-1 vi-1,...,vr-1 = P n-r+1 vi,...,vr-1 . Now, taking this equality for i = r -1 yields

which is equal to -S r by Claim 2. This proves Claim 3.

Since S 0 + • • • + S r is equal to (109), we have finally shown that whatever the value of c, ganit(poc)•N is circ-neutral, completing the proof of Proposition 30.