Kashiwara-Vergne and dihedral bigraded Lie algebras in mould theory

We introduce the Kashiwara-Vergne bigraded Lie algebra associated with a finite abelian group and give its mould theoretic reformulation. By using the mould theory, we show that it includes Goncharov's dihedral Lie algebra, which generalizes the result of Raphael and Schneps.

Kashiwara-Vergne Lie algebra krv • is the filtered graded Lie algebra introduced in [AT] and [AET]. It acts on the set of solutions of 'a formal version' of Kashiwara-Vergne conjecture. Related to conjectures on mixed Tate motives, it is expected to be isomorphic to the motivic Lie algebra (cf. [F]).
A bigraded variant lkrv •• of krv • is introduced in [RS] where they give its interpretation in terms of Ecalle's mould theory ([Ec81,Ec03,Ec11]). The results in [M, RS] give an inclusion of bigraded Lie algebras Our objective of this paper is to extend it to any Γ by exploiting Ecalle's mould theory with self-contained proofs. Our results are exhibited as follows: (i) In Definition 2.1, we introduce the filtered graded Q-linear space krv(Γ) • which generalizes krv • . In Theorem 2.15, we show that krv(Γ) • is identified with the Q-linear space of finite polynomial-valued alternal moulds satisfying Ecalle's senary relation (2.14) and whose swap's (1.8) are pus-neutral (1.9), that is, there is an isomorphism of Q-linear spaces   which extends (0.1). By imposing the distribution relation there the inclusion Fil 2 D D(Γ) •• ֒→ lkrvd(Γ) •• is similarly obtained in Corollary 3.15. In Appendix A, we give self-contained proofs of several fundamental properties on the ari-bracket of the Lie algebra ARI(Γ) of moulds associated with Γ. In Appendix B, we discuss moulds arising from the multiple polylogarithms evaluated at roots of unity.
Acknowledgements. We thank for L. Schneps who gave comments on the first version of the paper and informing us [RS]. We are grateful to the referee whose comments helped us to improve the paper in a better form. H.F. and N.K. have been supported by grants JSPS KAKENHI JP18H01110 and JP18J14774 respectively.

Preparation on mould theory
We prepare several techniques of moulds which will be employed in our later sections. The notion of moulds, the alternality, flexions and the ari-bracket associated with a finite abelian group Γ are explained in §1.1 and §1.2. In §1.3, we explain that the set ARI(Γ) al/al of bialternal moulds forms a Lie algebra under the ari-bracket (whose self-contained proof is given in Appendix A). In §1.4, we introduce the set ARI push/pusnu (Γ) of push-invariant and pus-neutral moulds and show that it forms a Lie algebra under the ari-bracket in Theorem 1.32.
1.1. Moulds and alternality. We introduce and discuss moulds associated with a finite abelian group Γ.
The notion of moulds was invented by Ecalle (cf. [Ec81,). For our convenience we employ the following formulation influenced by [Sch15] which is different from the one employed in [C,Définition 1 or Définition II.1] and [Sau,§4.1].
We prepare the following algebraic formulation which is useful to present the notion of the alternality of mould: Put X := xi σ i∈N,σ∈Γ . Let X Z be the set such that X Z := {( u σ ) | u = a 1 x 1 + · · · + a k x k , k ∈ N, a j ∈ Z, σ ∈ Γ} , and let X • Z be the non-commutative free monoid generated by all elements of X Z with the empty word ∅ as the unit. Occasionally we denote each element ω = u 1 · · · u m ∈ X • Z with u 1 , . . . , u m ∈ X Z by ω = (u 1 , . . . , u m ) as a sequence. The length of ω = u 1 · · · u m is defined to be l(ω) := m. where α and β run over X • Z . We set A X := Q X Z to be the non-commutative polynomial algebra generated by X Z (i.e. A X is the Q-linear space generated by X • Z ). 2 We equip A X with a product X : A ⊗2 X → A X (called the shuffle product) which is linearly defined by ∅ X ω := ω X ∅ := w and (1.1) uω X vη := u(ω X vη) + v(uω X η), for u, v ∈ X Z and ω, η ∈ X • Z . Then the pair (A X , X) forms a commutative, associative, unital Q-algebra.
We encode it with the induced depth filtrations. We exhibit a couple of examples of alternal moulds for Γ = {e} below.
Therefore, we obtain Hence, M f is an alternal mould. (b). The proof is given in [C,Lemme II.5], but we prove the alternality of A by using the following lemma: Lemma 1.5. For r 2 and s 0, we have where ω i := ( xi e ) for i ∈ N. Proof. When s = 0, the claim is clear. Assume s 1. We prove by induction on r + s. By the definition of the mould A, we easily see the case of (r, s) = (2, 1). For r 2 and s 1, we have (ω 1 , . . . , ω r−1 ); (ω r+1 , . . . , ω r+s−1 ) α å A(α, ω r+s ).
Hence, the mould A is alternal.
Remark 1.6. Assume that u 1 , . . . , u m ∈ F are algebraically independent over Q. Note that they are all Q-linear endomorphisms on ARI(Γ). We remark that neg • neg = id and mantar • mantar = id.
Example 1.9. By direct computations we observe that the mould P ∈ ARI({e}) defined by is push-invariant, but is not alternal.
Proof. Let m 0. We have By applying Lemma A.2.
(1) for c = ∅, f = ∅ to the first term and by applying Lemma A.2.
(2) for a = ∅, d = ∅ to the second term, we have By applying Lemma A.1 to the second term, we get A(b⌋ c1c2 ) = A(b⌋ c1 ). Similarly, for the third term, we get A( a1a2 ⌊b) = A( a2 ⌊b). Therefore, we calculate Hence, we obtain the claim.
We define the following bracket as with the bracket ari introduced in [Ec11, (2.40)].
We note that the bracket ari u (A, B) in the case when Γ = {e} also appears in [R00, (A.3)] and is denoted by [A, B] ari .
The following is also stated for moulds and bimoulds in [Sch15, Proposition 2.2.2] where her key formula (2.2.10) looks unproven and containing a signature error.
Proposition 1.14. The Q-linear space ARI(Γ) forms a filtered Lie algebra under the ari u -bracket. Proof. We give a self-contained proof in Appendix A.1.
The following proposition for Γ = {e} is shown in [SaSch, Appendix A].
Proposition 1.15. The Q-linear space ARI(Γ) al forms a filtered Lie subalgebra of ARI(Γ) under the ari u -bracket.
Proof. We prove this in Appendix A.2.
1.3. Swap and bialternality. We encode ARI(Γ) with another Lie algebra structure introduced by ari v . We prepare ARI(Γ), a copy of ARI(Γ). We denote Similarly to our previous sections, we work over the following algebraic formulation: . Let Y Z be the set such that and let Y • Z be the non-commutative free monoid generated by all elements of Y Z with the empty word ∅ as the unit. We set A Y := Q Y Z to be the non-commutative polynomial algebra generated by Y Z . In the same way to A X , it is equipped with a structure of a commutative, associative, unital Q-algebra with the shuffle product The flexions are also introduced in this setting.
Similarly to Lemma 1.12, the following holds.
Lemma 1.18. For any A ∈ ARI(Γ), arit v (A) forms a derivation of ARI(Γ) with respect to the product ×.
Proof. The arit v -bracket can be expressed by the exactly same formula as the arit ubracket in terms of flexions. Therefore it can be proved in the same way to the one of Lemma 1.12.
Proposition 1.20. The Q-linear space ARI(Γ) forms a filtered Lie algebra under the ari v -bracket.
Proof. It can be also proved in the same way to the one of Proposition 1.14.
In the same way to Definition 1.3, alternal moulds in ARI(Γ) can be introduced and we denote ARI(Γ) al to be its subset consisting of alternal moulds. Similarly to Proposition 1.15, the following holds.
Proposition 1.21. ARI(Γ) al forms a Lie algebra under the ari v -bracket.
Proof. It can be proved in the same way to the one of Proposition 1.15.
We define the Q-linear map swap : Definition 1.22 (cf. [Ec03], [Ec11]). The subset ARI(Γ) al/al of bialternal moulds is defined to be Here are observations on Example 1.4. 8 The lower suffix v is reflected by the notion of v-moulds in [Sch15]. 9 In [RS], the brackets aritv(A, B) and ariv(A, B) are denoted by arit and ari respectively.
and we get Hence, swap(A) is not alternal, which says A ∈ ARI(Γ) al/al .
Proposition 1.24. The Q-linear space ARI(Γ) al/al forms a filtered Lie subalgebra of ARI(Γ) al under the ari u -bracket.
Proof. We prove this in Appendix A.3.
Hence, the mould Q is pus-neutral.
Firstly we show that the set ARI push forms a Lie algebra under the ari u -bracket which was stated in [Ec11, §2.5] without a detailed proof. Proof. Let m 1. We put ω = u1, ..., um σ1, ..., σm . We have So we get Therefore, by using (1.10) and (1.11), we have Remark 1.29. This proposition improves the results of [RS,§4.1.3], which shows that the intersection ARI push (Γ) ∩ ARI pol al (Γ) forms a Lie algebra under the ari ubracket when Γ = {e}.
Secondly we show that the set of pus-neutral moulds is closed under the ari vbracket. The following proposition is a generalization of [RS,Lemma 21] which treats the case when Γ = {e}.
Proof. The proof goes in the same way to that of [RS]. Let m 1. Since the algebra ARI(Γ) is graded by depth, it is enough to prove A and B ∈ ARI(Γ) with depth k and l ∈ N with m = k + l.
Thus we complete the proof because ari We need the following lemma for the proof of Theorem 1.32.
Proof. This follows from the proof of [Sch15,Lemma 2.4.1], which actually works for BARI.
By Proposition 1.30, the right hand side is equal to 0. Hence swap(ari u (A, B)) is pus-neutral. Thus we obtain ari u (A, B) ∈ ARI(Γ) push/pusnu .
and also the map m N : ARI(Γ) → ARI(Γ N ) which is given by We define the following Q-linear subspaces which are subject to the distribution relations: Proof. First we prove that m N is a Lie algebra homomorphism, that is, Similarly, we have Here, we put S k,l := {i ∈ N | 1 i k − 1} ∪ {i ∈ N | l + 1 i m} for 1 k l m. Then we can divide the set of variables of the summation in the following : So by substituting ν i := τ i for i ∈ S k,l and ν i := τ i τ l+1 for k i l for the first summation, and by substituting ν i := τ i for i ∈ S k,l and ν i := τ i τ k−1 for k i l for the second summation, we calculate Hence we get Therefore, m N is a Lie algebra homomorphism. It is obvious that the map i N is a Lie algebra homomorphism. Since both m N and i N are Lie algebra homomorphisms, ker(i N − m N ) forms Lie algebra.
Corollary 1.36. The Q-linear subspace ARID(Γ) al/al forms a filtered Lie subalgebra of ARI(Γ) under the ari u -bracket.
Proof. It is a direct consequence of Proposition 1.24 and Proposition 1.35.

Kashiwara-Vergne Lie algebra
We introduce the Kashiwara-Vergne bigraded Lie algebra lkrv(Γ) •• associated with a finite abelian group Γ and give its mould theoretical interpretation by using ARI(Γ) push/pusnu . 2.1. Γ-variant of the KV condition. We investigate a variant of the defining conditions of Kashiwara-Vergne graded Lie algebra associated with a finite abelian group Γ (cf. Definition 2.1) and explain its mould theoretical interpretation in Theorem 2.15.
Let L = ⊕ w 1 L w be the free graded Lie Q-algebra generated by N + 1 variables x and y σ (σ ∈ Γ) with deg x = deg y σ = 1. Here L w is the Q-linear space generated by Lie monomials whose total degree is w. Occasionally we regard L as a bigraded Lie algebra L •• = ⊕ w,d L w,d , where L w,d is the Q-linear space generated by Lie monomials whose weight (the total degree) is w and depth (the degree with respect to all y σ ) is d. We encode L with a structure of filtered graded Lie algebra by the filtration Fil d D L w := ⊕ N d L w,N for d > 0 and denote the associated bigraded Lie algebra by gr D is regarded as the universal enveloping algebra of L and is encoded with the induced degree. Similarly A is encoded with a structure of bigraded algebra; we also encode A with a structure of filtered graded algebra. We define the action of τ ∈ Γ on A (hence on L) by τ (x) = x and τ (y σ ) = y τ σ .
For any h ∈ A, we denote We denote π Y to be the composition of the natural projection and inclusion: We put Cyc(A) to be Q-linear space generated by cyclic words of A and tr : A ։ Cyc(A) to be the trace map, the natural projection to Cyc(A) (cf. [AT]).
We note that such G = G(F ) uniquely exists when w > 1. For d 1, we put whence we get the claim.
We start with the following technical lemma which is required to our later arguments.  Definition 2.6. Let tder be the set of tangential derivation of L, the derivation D {Fσ}σ,G of L defined by x → [x, G] and y σ → [y σ , F σ ] for some F σ , G ∈ L. It forms a Lie algebra by the bracket (2.5) . The action of Γ on L induces the Γ-action on tder. We denote its invariant part by tder Γ . We mean sder to be the set of special derivations, tangential derivations such that D {Fσ },G (x + σ y σ ) = 0 and sder Γ to be its intersection with tder Γ , both of which forms a Lie subalgebra of tder. We put mt := ⊕ (w,d) =(1,0) L w,d . It forms a Lie algebra by the bracket We occasionally regard mt as a Lie subalgebra of tder by f → D {σ(f )},0 . We note that the condition (KV1) is equivalent to D {σ(F )},G ∈ sder Γ .
We regard sder Γ and mt as filtered graded Lie algebras by encoding them with The following is required in the next section.
Lemma 2.7. We have a natural graded Lie algebra homomorphism D 2 ]. Since it belongs to sder Γ , D 3 is expressed as D {τ (F3)},G3 for some F 3 ∈ L and G 3 ∈ L. By whereF 1 ∈ gr d1 D L w1 andF 2 ∈ gr d2 D L w2 are the residue classes of F 1 and F 2 . Therefore our map is a Lie algebra homomorphism.
Proposition 2.8. The map ma sending h → ma h induces a filtered graded Lie algebra isomorphism Proof. It can be proved completely in a same way to that of [Sch15,Theorem 3.4.2].
We prepare the following technical lemma which is required to the proof of a reformulation of (KV1) in Lemma 2.10.
Lemma 2.9. Let H ∈ L w with w 2. Assume that H has no words starting with any y σ and ending in any y τ . Then there exists G ∈ L w−1 such that H = [x, G].
Proof. The proof goes on the same way to the proof of [Sch12, Proposition 2.2]. Define the derivation ∂ x of A sending x → 1 and y σ → 0 and the Q-linear endomorphism sec of A by sec(h) : Then by our assumption, we have P ∈ A such that xP = σ H yσ y σ . Then by [R02] Proposition 4.2.2 and ∂ i It remains to show that G = sec(P ) is in L, which follows from exactly the same arguments to the last half of the proof of [Sch12, Proposition 2.2].
Lemma 2.10. Let F ∈ L w with w 1. Then saying (KV1) for F is equivalent to saying that So H has no words starting with any y σ and ending in any y τ . By (2.12), Then H has no words starting with any y σ and ending in any y τ . By Lemma 2.9, there is a G ∈ L w such that H = [G, x]. Whence we get (KV1). Secondly, by α(F ) ∈ L w , we have α(F ) y β = (−1) w−1 anti(α(F ) y β ). Therefore α(F ) y β = β(F ) yα is equivalent to α(F ) y β = (−1) w−1 anti(β(F ) yα ).
The following reformulation is suggested by the arguments in [Sch12, Appendix A].
Next we consider the condition (KV2).
So we get the claim.
The following definition of the mould version of krv • is suggested by our previous lemmas.
Definition 2.14. We define the Q-linear space ARI(Γ) sena/pusnu to be the subset of moulds M in ARI which satisfy the senary relation (2.14) and whose swap satisfy the pus-neutrality (2.27).
Theorem 2.15. The map sending F ∈ A → maf ∈ M(F ; Γ) induces an isomorphism of filtered Q-linear spaces Proof. The restriction of our map decomposes as By Proposition 2.8, we see that it gives an isomorphism mt ≃ ARI(Γ) fin,pol al as (actually filtered) Q-linear spaces. Thus we get the claim by our previous lemmas.
The authors are not sure if the bigger space ARI(Γ) sena/pusnu is equipped with a structure of Lie algebra under the ari u -bracket or not although we show that a related space ARI(Γ) push/pusnu forms a Lie algebra in Theorem 1.32.

2.2.
Kashiwara-Vergne bigraded Lie algebra. Based on our arguments in the previous subsection, we introduce a Γ-variant lkrv(Γ) •• of the Kashiwara-Vergne bigraded Lie algebra lkrv ( [RS]) in Definition 2.17 and give a mould theoretical interpretation in Theorem 2.22. As a corollary we show that it forms a Lie algebra in Theorem 2.23.
Definition 2.17. Kashiwara-Vergne bigraded Lie algebra is defined to be the bigraded Q-linear space lkrv(Γ) •• = ⊕ w>1,d>0 lkrv(Γ) w,d , where lkrv(Γ) w,d is the Qlinear space consisting ofF ∈ gr d D L w whose lift F ∈ Fil d D L w satisfies the following relations, 'the leading-terms' of (KV1) and (KV2), We note that such G ∈ L w is in Fil d+1 D L w and is uniquely determined modulo Fil d+2 D L w by (LKV1). We note that, by (LKV2) and dim L w,1 = 1, we have lkrv w,d = {0} for d = 1. which generalizes [RS,Proposition 2], that is, the associated graded Q-linear space gr D krv(Γ) • of the filtered Lie algebra krv(Γ) • is embedded to lkrv(Γ) •• . We do not know if it is an isomorphism.
For F ∈ Fil d D L w , we put f to be the element in L w,d corresponding to (−1) w−dF ∈ gr m D L w under the natural identification gr d D L w ≃ L w,d . By abuse of notation, (2.32) f = (−1) w−dF .
We write f = f x x + σ f yσ y σ . We also put and writef =f x x + σf yσ y σ . The following is a bigraded variant of Lemma 2.10.
Proof. The proof goes similarly to Lemma 2.10.
The following might be regarded as a bigraded variant of Lemma 2.11. Lemma 2.20. Letf ∈ L w,d with w > 1. Then (2.33) for any γ ∈ Γ is equivalent to push-invariance (1.5) for M = maf .
Proof. The proof goes similarly to Lemma 2.11. The condition (2.33) for γ ∈ Γ is reformulated to Proof. We note that actually only the terms for r = d contributes in the above equation. Decomposef as in (2.2). Then by the arguments in the proof of Lemma 2.13, (2.27) is equivalent to for all (σ 1 , . . . , σ r ) ∈ Γ r and (e 0 , . . . , e r−1 , 0) ∈ E r w . It is nothing but which is equivalent to (LKV2). Proof. Our map decomposes as The first map is injective and the second one is isomorphic by Proposition 2.8. Our claim follows by our previous lemmas.
As a generalization of [RS,Proposition 1], we obtain the following.
Proof. As for the morphism (2.35), the second map is Lie algebra homomorphism by Proposition 2.8. It is easy to see that the first map forms a Lie algebra homomorphism when we encode lkrv(Γ) •• with the bracket (2.6). Thus our claim follows because it is shown that ARI(Γ) push/pusnu forms a Lie algebra by Theorem 1.32 and so ARI(Γ) al does by Lemma 1.15.
Similarly to Definition 1.34, we impose a distribution relation on lkrv(Γ).
Definition 2.25. For N 1 and Γ N = {g N ∈ Γ | g ∈ Γ}, we consider the map We define the following Q-linear subspace As a corollary of Theorems 2.22 and 2.23, we obtain the following corollary.
Corollary 2.26. The space lkrvd(Γ) forms a bigraded Lie algebra and we have the following isomorphism of bigraded Lie algebras Proof. By definition we see that both i N and m N form Lie algebra homomorphisms and by Proposition 1.14 ARI(Γ) forms a Lie algebra. Therefore our claim follows from the commutativity of the following diagrams and Proposition 1.35:
The following reformulation of the above dihedral symmetry relations is useful in our later arguments.
Thus we see that (3.5) is equivalent to the condition for swap(M Z ∼ ) being in ARI(Γ) al . Therefore our map forms a Q-linear isomorphism (3.12) by Theorem 3.2.
Since ARI(Γ) al/al , and hence the right hand side of (3.12), forms a Lie algebra by Proposition 1.24, we learn that Fil 2 Corollary 3.6. The map M in Theorem 3.5 induces a Lie algebra isomorphism between (3.14) Here the left hand side means the depth>1-part of D(Γ) •• and the right hand side means the finite polynomial-valued part of the subset of ARID(Γ) al/al (cf. Definition 1.34) consisting of M with depth>1.
Proof. Since the distribution relation (3.7) is equivalent to Since ARID(Γ) al/al , and hence the right hand side of (3.14), forms a Lie algebra by Corollary 1.36, and Fil 2 D D(Γ) •• forms a Lie subalgebra of D(Γ) •• by [G01a, Theorem 5.2], our claim follows.
The following three relations should be called as the cyclic symmetry relation, the inversion relation and the reflection relation respectively (compare them with (3.9), (3.10) and (3.11) with m + 1 replaced with m).
The following can be also found in [Sch15,Lemma 2.5.5] but we give a different proof below.
Theorem 3.13. There is an embedding Therefore, we obtain M ∈ ARI(Γ) push/pusnu . To see that it is a Lie algebra homomorphism is immediate.
As a corollary, by taking an intersection with ARI(Γ) fin,pol al in the embedding of the above theorem, we obtain the following inclusion which generalizes [RS,Theorem 3].
By imposing the distribution relation, we also obtain the following.  Proof. It follows from Corollary 2.26, Corollary 3.6 and Theorem 3.13.
It looks interesting to see if the map is an isomorphism.
Remark 3.16. By (3.12), Theorem 2.22, Theorem 3.13 and Corollary 3.14, we obtain the commutative diagram (3.24) of Lie algebras: On the ari-bracket of ARI(Γ) In this appendix, we give self-contained proofs of fundamental properties of the ari-bracket of ARI(Γ), that is, Proposition 1.14, 1.15 and 1.24, which are required in this paper.
A.1. Proof of Proposition 1.14. We give a proof Proposition 1.14 which claims that ARI(Γ) forms a Lie algebra, by showing that it actually forms a pre-Lie algebra.
We start with three fundamental lemmas which can be proved directly by simple computations.
All the other cases can shown similarly.
Proof. We present a proof for (1). When a b ⌈c = def , the following depict all the possible cases of the positions of a, b ⌈c, d, e and f .
The first, the second and the third cases correspond (I), (II) and (III) in (1) respectively 16 . The claim for (2) can be proved in the same way.
The following formula is essential to prove Proposition 1.14. It is stated in [Sch15,(2.2.10)] (where it looks that there is an error on the signature) without a proof.
Proposition A.4. For any A, B ∈ ARI(Γ), we have Proof. Let m 0. Then we have By using Lemma A.2, we have   Here, for the first term, we have By using Lemma A.1, we get Therefore, cancellation 20 yields Here, the first term is calculated such that Then we have ari u (A, B) = preari u (A, B) − preari u (B, A).
Proof. We have Therefore, by using Proposition A.4, we obtain the claim.
By Proposition A.6, it is equal to 0. So we obtain the Jacobi identity.
By using Lemma A.7 for r = 2, we get By using alternality of A, we calculate By using alternality of B, we obtain
Here, Sh ω ′ 2 ;η ′ 2 x = 0 holds for x ∈ X Z if and only if (ω ′ 2 , η ′ 2 ) = (x, ∅) or (∅, x). So we get particular, the coefficient of x nr −1 y ǫr x nr−1−1 y ǫr−1ǫr · · · x n1−1 y ǫ1···ǫr is (−1) r Li X n1,...,nr (ǫ 1 , . . . , ǫ r ). Definition 2.4 enables us to calculate a relation between the associator Φ N KZ and the mould Zag as follows ma ( Li n (1)y n 1´∈ÂC and define (B.4) Φ N KZ, * := Φ KZ,corr · q(π Y ( Φ N KZ )) (for π Y and q, see §2.1). It is shown in [R02] that Φ N KZ is group-like with respect to the shuffle (deconcatenation) coproduct ofÂ C and Φ N KZ, * is so with respect to the harmonic coproduct of Im π Y . They correspond to the shuffle and the harmonic product among multiple L-values respectively. The regularized double shuffle relations (the shuffle and the harmonic products and the regularization relations (B.4)) are the defining equations of his torsor DMR µ 22 for µ ∈ C × which contains Φ N KZ as a specific point when µ = 2π √ −1. It is equipped with a free and transitive action of the group DMR 0 . Its associated Lie algebra, which he denotes by dmr 0 , is a filtered graded Lie algebra defined by the regularized double shuffle relations modulo products. Our dihedral Lie algebra D(µ N ) •• should be called its bigraded variant, defined by 'their highest depth parts' of the relations. By definition, it contains the associated graded grdmr 0 . By translating Ecalle's pictures including the diagram (B.2) to this setting, we might learn more enriched structures on these Lie algebras.