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ACPMS On Kashiwara-Vergne bigraded Lie algebra in mould theory

Nao Komiyama

Nagoya University

27/11/2020

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Contents

1

Introduction Kashiwara-Vergne group Kashiwara-Vergne Lie algebra

2

Mould Definition of mould Lie algebra ARI(Γ) of moulds

3

Lie algebra Lie subalgebras of ARI(Γ) Kashiwara-Vergne bigraded Lie algebra lkrv(Γ)••

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Section 1 Introduction

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Introduction ∼Kashiwara-Vergne group∼

Kashiwara and Vergne proposed a conjecture related to the Campbell-Baker-Hausdorff series in the following paper: The Campbell-Hausdorff formula and invariant hyperfunctions,

• Invent. Math. 47 (1978), no. 3, 249-272.

There is one of the formulations of the above KV conjecture by Alekseev, Enriquez and Torossian.

Notation

• Let F2 be the completed free Lie algebra with two variables X0 and X1.
• Denote by UF2 = k⟨⟨X0, X1⟩⟩ the non-commutative formal power series

ring defined as the universal enveloping algebra of F2.

• Put exp F2 to be the image of F2 under the map exp : F2 → UF2 defined

by exp(x) := ∑

n≥0

xn n! (x ∈ F2).

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Introduction ∼Kashiwara-Vergne group∼

Generalized Kashiwara-Vergne problem ([Alekseev-Enriquez-Torossian])

Find a group automorphism P : exp F2 → exp F2 such that P ∈ TAut exp F2 and P satisfies P(eX0eX1) = eX0+X1 and the coboundary Jacobian condition δ ◦ J(P) = 0. Here, P ∈ TAut exp F2 means that P is in Aut exp F2 such that P(eX0) = p1eX0p−1

1

and P(eX1) = p2eX1p−1

2

for some p1, p2 ∈ exp F2. J stands for the Jacobian cocycle and δ means the differential map.

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Introduction ∼Kashiwara-Vergne group∼

Definition ([A-E-T; 2010], [A-T; 2012])

The Kashiwara-Vergne group KRV is defined to be the set of P ∈ TAut exp F2 which satisfies P(eX0eX1) = eX0+X1 and the coboundary Jacobian condition δ ◦ J(P) = 0.

Remark

The set KRV forms a subgroup of Aut exp F2.

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Introduction ∼Kashiwara-Vergne group∼

We denote KRV0 to be a subgroup of KRV consisting of P without linear terms in p1 and p2. There are the following inclusions.

Theorem ([A-E-T; 2010], [A-T; 2012])

GRT1 ⊂ KRV0.

Theorem ([Schneps; 2012])

DMR0 ⊂ KRV0. Here, GRT1 is the Grothendieck-Teichm¨ uller group introduced by Drinfel’d, and DMR0 is the double shuffle group introduced by Racinet. cf.) H. Furusho, Around associators. Automorphic forms and Galois representations, (2014) Vol. 2, 105–117.

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Introduction ∼Kashiwara-Vergne Lie algebra∼

To recall the definition of KV Lie algebra, we prepare some notations.

• Let L = ⊕w≥1Lw be the free graded Lie Q-algebra generated by two

variables x and y with deg x = deg y = 1 (Lw is the Q-linear space generated by Lie monomials whose total degree is w).

• The non-commutative polynomial algebra

A = Q⟨x, y⟩ is regarded as the universal enveloping algebra of L.

• Put Cyc(A) to be the Q-linear space generated by cyclic words of A,

and put the trace map tr : A ↠ Cyc(A) to be the natural projection.

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Introduction ∼Kashiwara-Vergne Lie algebra∼

Definition ([A-E-T; 2010], [A-T; 2012])

The Kashiwara-Vergne (graded) Lie algebra is the graded Q-linear space krv• = ⊕w≥2krvw. Here, its degree w-part krvw is defined to be the set of Lie elements F ∈ Lw such that there exists G = G(F) in Lw with [x, G] + [y, F] = 0 (KV1) and α ∈ Q with tr(Gxx + Fyy) = α · tr ((x + y)w − xw − yw) (KV2) when we write F = Fxx + Fyy and G = Gxx + Gyy in A. We note that such G = G(F) uniquely exists for F ∈ Lw when w ≥ 2.

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Introduction ∼Kashiwara-Vergne Lie algebra∼

The Lie algebra structure of krv•

• Let tder be the set of tangential derivation of L.

The derivation DF,G of L defined by x → [x, G] and y → [y, F] for some F, G ∈ L. It forms a Lie algebra by the bracket [DF1,G1, DF2,G2] = DF1,G1 ◦ DF2,G2 − DF2,G2 ◦ DF1,G1.

• We denote sder to be set of special derivations, which are tangential

derivations D such that D(x + y) = 0. It forms a Lie subalgebra of tder. The Lie algebra structure of krv• is defined to be compatible with that of sder under the embedding F ∈ krv• to DF,G(F) ∈ sder.

Today’s goal

To give a Kashiwara-Vergne bigraded Lie algebra lkrv(Γ)••.

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Section 2 Mould

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Mould ∼Definition∼

Early 1980s, the mould was introduced by Ecalle in his paper: Les fonctions r´ esurgentes, Tome I, II, III. In 2000s, he applied the moulds to study of multiple zeta values in papers:

• ARI/GARI, la dimorphie et l’arithm´

etique des multizetas,

• The flexion structure and dimorphy: flexion units, singulators, generators,

and the enumeration of multizeta irreducibles. cf.) Schneps, ARI, GARI, Zig and Zag: An introduction to Ecalle’s theory

• f multiple zeta values, arXiv:1507.01534.

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Mould ∼Definition∼

Let Γ be a finite abelian group. We set F := ∪

m⩾1

Q(x1, . . . , xm).

Definition ([Furusho-K, Definition 1.1])

A mould on Z⩾0 with values in F is a collection (a sequence) M = (Mm(x1, . . . , xm))m∈Z⩾0 = ( M0(∅), M1(x1), M2(x1, x2), . . . ) , with M0(∅) ∈ Q and Mm(x1, . . . , xm) ∈ Q(x1, . . . , xm)⊕Γ⊕m for m ⩾ 1, which is described by a summation Mm(x1, . . . , xm) = ⊕

(σ1,...,σm)∈Γ⊕mMm σ1,...,σm(x1, . . . , xm)

where each Mm

σ1,...,σm(x1, . . . , xm) ∈ Q(x1, . . . , xm).

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Mould ∼Definition∼

• We denote the set of all moulds with values in F by M(F; Γ).

The set M(F; Γ) forms a Q-linear space by A + B := (Am(x1, . . . , xm) + Bm(x1, . . . , xm))m∈Z⩾0, cA := (cAm(x1, . . . , xm))m∈Z⩾0, for A, B ∈ M(F; Γ) and c ∈ Q.

• We define a product on M(F; Γ) by

(A × B)m

σ1,...,σm(x1, . . . , xm) := m

∑

i=0

Ai

σ1,...,σi(x1, . . . , xi)Bm−i σi+1,...,σm(xi+1, . . . , xm),

for A, B ∈ M(F; Γ) and for m ⩾ 0 and for (σ1, . . . , σm) ∈ Γ⊕m.

• Then the pair (M(F; Γ), ×) is a non-commutative, associative, unital

Q-algebra. Here, the unit I ∈ M(F; Γ) is given by I := (1, 0, 0, . . . ).

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Mould ∼Example∼

We give some examples.

Example

(1). Define A ∈ M(F; Γ) by

Am

σ1,...,σm(x1, . . . , xm) :=

{ 0 (m = 0, 1),

1 (x2−x1) · · · 1 (xr −xr−1)

(m ≥ 2),

for m ≥ 0 and for σ1, . . . , σm ∈ Γ. (2). Define B ∈ M(F; Γ) by

Bm

σ1,...,σm(x1, . . . , xm) :=

{ 0 (m = 0, 1),

1 x1 + · · · + 1 xm − 1 x1+···+xm

(m ≥ 2),

for m ≥ 0 and for σ1, . . . , σm ∈ Γ.

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Mould ∼Lie algebra ARI(Γ)∼

• Put ARI(Γ) := {M ∈ M(F; Γ) | M0(∅) = 0}

(this set forms a non-unital subalgebra of (M(F; Γ), ×)). For M ∈ ARI(Γ), we sometimes denote Mm

σ1,...,σm(x1, . . . , xm) by

Mm (x1, ..., xm

σ1, ..., σm

) .

• In order to give a Lie algebraic structure to ARI(Γ), we prepare the

algebraic formulation: Put X := {(xi

σ

)}

i∈N,σ∈Γ. Let XZ be the set such that

XZ := {(u

σ) | u = a1x1 + · · · + akxk, k ∈ N, aj ∈ Z, σ ∈ Γ} .

Let X •

Z be the non-commutative free monoid generated by all elements of

XZ with the empty word ∅ as the unit. Occasionally we denote each element ω = u1 · · · um ∈ X •

Z with u1, . . . , um ∈ XZ by ω = (u1, . . . , um) as

a sequence. The length of ω = u1 · · · um is defined to be l(ω) := m.

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Mould ∼Lie algebra ARI(Γ)∼

Definition ([F-K, Definition 1.8])

The flexions are the four binary operators

∗⌈∗, ∗⌉∗, ∗⌊∗, ∗⌋∗ : X •

Z × X • Z → X • Z which are defined by

β⌈α :=

(

b1 + · · · + bn + a1, a2, . . . , am σ1, σ2, . . . , σm

) , α⌉β := (

a1, . . . , am−1, am + b1 + · · · + bn σ1, . . . , σm−1, σm

) ,

β⌊α :=

(

a1, . . . , am τ −1

n

σ1, . . . , τ −1

n

σm

) , α⌋β := (

a1, . . . , am σ1τ −1

1

, . . . , σmτ −1

1

) ,

∅⌈γ := γ⌉∅ := ∅⌊γ := γ⌋∅ := γ, γ⌈∅ := ∅⌉γ := γ⌊∅ := ∅⌋γ := ∅,

for α = (a1,...,am

σ1,...,σm

) , β = (

b1,...,bn τ1,...,τn

) ∈ X •

Z (m, n ⩾ 1) and γ ∈ X • Z.

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Mould ∼Lie algebra ARI(Γ)∼

Put x0 := ∅ and xm := (x1, ..., xm

σ1, ..., σm

) for m ⩾ 1.

Definition ([F-K, Definition 1.9]

• cf. [Ecalle; 2011], [Schneps; 2015])

Let B ∈ ARI(Γ). The linear map aritu(B) : ARI(Γ) → ARI(Γ) is defined by (aritu(B)(A))0(x0) := (aritu(B)(A))1(x1) := 0 and for m ≥ 2 (aritu(B)(A))m(xm) := ∑

xm=αβγ β,γ̸=∅

Al(α,γ)(αβ⌈γ)Bl(β)(β⌋γ) − ∑

xm=αβγ α,β̸=∅

Al(α,γ)(α⌉βγ)Bl(β)(α⌊β), for A ∈ ARI(Γ).

Remark

Assume that u1, . . . , um ∈ F are algebraically independent over Q. For M ∈ ARI(Γ), we denote Mm (u1, ..., um

σ1, ..., σm

) to be the image of Mm (x1, ..., xm

σ1, ..., σm

) under the field embedding Q(x1, . . . , xm) ֒ → F sending xi → ui.

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Mould ∼Lie algebra ARI(Γ)∼

Example

For m = 2, we have

(aritu(B)(A))2(x2) = A1 (x1 + x2 σ2 ) B1 ( x1 σ1σ−1

2

) − A1 (x1 + x2 σ1 ) B1 ( x2 σ2σ−1

1

) ,

and for m = 3, we have

(aritu(B)(A))3(x3) = A2 (x1, x2 + x3 σ1, σ3 ) B1 ( x2 σ2σ−1

3

) + A1 (x1 + x2 + x3 σ3 ) B2 ( x1, x2 σ1σ−1

3 , σ2σ−1 3

) + A2 (x1 + x2, x3 σ2, σ3 ) B1 ( x1 σ1σ−1

2

) − A2 (x1 + x2, x3 σ1, σ3 ) B1 ( x2 σ2σ−1

1

) − A1 (x1 + x2 + x3 σ1 ) B2 ( x2, x3 σ2σ−1

1 , σ3σ−1 1

) − A2 (x1, x2 + x3 σ1, σ2 ) B1 ( x3 σ3σ−1

2

) .

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Mould ∼Lie algebra ARI(Γ)∼

Lemma ([F-K, Lemma 1.10])

For any A ∈ ARI(Γ), aritu(A) forms a derivation of ARI(Γ) with respect to the product ×, that is, for any B, C ∈ ARI(Γ), we have aritu(A)(B × C) = aritu(A)(B) × C + B × aritu(A)(C). The bilinear map ariu : ARI(Γ)⊗2 → ARI(Γ) is defined by ariu(A, B) := aritu(B)(A) − aritu(A)(B) + [A, B] for A, B ∈ ARI(Γ). Here, we have [A, B] := A × B − B × A.

Proposition ([F-K, Proposition 1.12])

The Q-linear space ARI(Γ) forms a Lie algebra under the ariu-bracket.

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Section 3 Lie algebra

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Lie subalgebra of ARI(Γ)

• We set AX := Q⟨XZ⟩ to be the non-commutative polynomial algebra

generated by XZ (i.e. AX is the Q-linear space generated by X •

Z).

• We equip AX a product : A⊗2

X → AX which is linearly defined by

∅ ω := ω ∅ := w and uω vη := u(ω vη) + v(uω η), for u, v ∈ XZ and ω, η ∈ X •

Z.

• Then the pair (AX, ) forms a commutative, associative, unital

Q-algebra. We define the family { Sh (ω;η

α

)}

ω,η,α∈X •

Z in Z by

ω η = ∑

α∈X •

Z

Sh (ω; η α ) α.

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Lie subalgebra of ARI(Γ)

Definition ([F-K, Definition 1.4])

A mould M ∈ ARI(Γ) is called alternal if ∑

α∈X •

Z

Sh ((x1, ..., xp

σ1, ..., σp

) ; (xp+1, ..., xp+q

σp+1, ..., σp+q

) α ) Mp+q(α) = 0, for all p, q ⩾ 1. The Q-linear space ARI(Γ)al is defined to be the subset of moulds M ∈ ARI(Γ) which are alternal.

Proposition ([F-K, Proposition 1.13])

The Q-linear space ARI(Γ)al forms a Lie subalgebra of ARI(Γ) under the ariu-bracket. cf.) The above proposition for Γ = {e} is shown in [SaSch, Appendix A].

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Lie subalgebra of ARI(Γ)

• We prepare ARI(Γ), a copy of ARI(Γ).
• We denote Mm

σ1,...,σm(x1, . . . , xm) by Mm (σ1, ..., σm x1, ..., xm ) for each element

M ∈ ARI(Γ) to distinguish it from an element in ARI(Γ). (For M ∈ ARI(Γ), Mm

σ1,...,σm(x1, . . . , xm) is denoted by Mm (x1, ..., xm σ1, ..., σm

) )

• We define the Q-linear map swap : ARI(Γ) → ARI(Γ) by

swap(M)m ( σ1, . . . , σm x1, . . . , xm ) = Mm ( xm, xm−1 − xm, . . . , x2 − x3, x1 − x2 σ1 · · · σm, σ1 · · · σm−1, . . . , σ1σ2, σ1 )

for any mould M = ( Mm (x1, ..., xm

σ1, ..., σm

)) ∈ ARI(Γ).

ARI(Γ)al/al := {M ∈ ARI(Γ)al | swap(M) ∈ ARI(Γ)al, M1 (x1

σ1

) = M1 (

−x1 σ−1

1

) }.

Proposition ([F-K, Proposition 1.21])

The Q-linear space ARI(Γ)al/al forms a Lie subalgebra of ARI(Γ)al under the ariu-bracket.

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Lie subalgebra of ARI(Γ)

• Define maps push : ARI(Γ) → ARI(Γ) and pus : ARI(Γ) → ARI(Γ) by

push(M)m(

x1, . . . , xm σ1, . . . , σm

) = Mm( −x1 − · · · − xm,

x1, . . . , xm−1 σ−1

m ,

σ1σ−1

m ,

. . . , σm−1σ−1

m

) , pus(N)m(

σ1, . . . , σm x1, . . . , xm

) = Nm(

σm, σ1, . . . , σm−1 xm, x1, . . . , xm−1

) , for M ∈ ARI(Γ) and N ∈ ARI(Γ).

• We call a mould M ∈ ARI(Γ) push invariant when we have

push(M) = M, and call a mould N ∈ ARI(Γ) pus-neutral when we have

m

∑

i=1

pusi(N)m (σ1, ..., σm

x1, ..., xm

)  = ∑

i∈Z/mZ

Nm (σi+1, ..., σi+m

xi+1, ..., xi+m

)   = 0, for all m ⩾ 1 and σ1, . . . , σm ∈ Γ.

• We define ARI(Γ)push/pusnu to be the set of moulds M ∈ ARI(Γ) which

is push-invariant and whose swap(M) ∈ ARI(Γ) is pus-neutral.

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Lie subalgebra of ARI(Γ)

Theorem ([F-K, Theorem 1.28])

The set ARI(Γ)push/pusnu forms a Lie subalgebra of ARI(Γ) under the ariu-bracket. The set ARI(Γ)al/al is encoded with the depth filtration {Film

DARI(Γ)al/al}m≥0 where Film DARI(Γ)al/al is the collection of moulds

M ∈ ARI(Γ)al/al with Mr(x1, . . . , xr) = 0 for r < m. The Lie algebra structure of ARI(Γ)al/al is compatible with the depth filtration.

Theorem ([F-K, Theorem 3.13])

There is an embedding Fil2

DARI(Γ)al/al ֒

→ ARI(Γ)push/pusnu

• f graded Lie algebras.

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Kashiwara-Vergne bigraded Lie algebra lkrv(Γ)••

• Let L = ⊕w≥1Lw be the free graded Lie Q-algebra generated by N + 1

variables x and yσ (σ ∈ Γ) with deg x = deg yσ = 1. Here Lw is the Q-linear space generated by Lie monomials whose total degree is w.

• The (N + 1)-variable non-commutative polynomial algebra

A = Q⟨x, yσ; σ ∈ Γ⟩ is regarded as the universal enveloping algebra of L.

• We define the action of τ ∈ Γ on A (hence of L) by

τ(x) = x and τ(yσ) = yτσ.

• Put πY to be the composition of the natural projection and inclusion:

πY : A ↠ A/A · x ≃ Q ⊕ (⊕σ∈ΓAyσ) ֒ → A, and q to be the Q-linear isomorphism on A defined by q(xe0yσ1xe1yσ2 · · · xer−1yσr xer ) = xe0yσ1xe1yσ2σ−1

1

· · · xer−1yσrσ−1

r−1xer . Nao KOMIYAMA ACPMS 27/11/2020 27 / 35

Kashiwara-Vergne bigraded Lie algebra lkrv(Γ)••

Definition (Kashiwara-Vergne bigraded Lie algebra: [F-K, Definition 2.17])

Define lkrv(Γ)•• = ⊕w>1,d>0lkrv(Γ)w,d to be the bigraded Q-linear space, where lkrv(Γ)w,d is the Q-linear space consisting of ¯ F ∈ grd

DLw whose lift

F ∈ Fild

DLw satisfies the following relations:

[x, G] + ∑

τ∈Γ

[yτ, τ(F)] ≡ 0 mod Fild+2

D

Lw+1, (LKV1) tr ◦ q ◦ πY (F(z; (yσ))) ≡ 0 mod tr(Fild+1

D

Aw) (LKV2) for a certain G = G(F) ∈ Lw. Here, we put z = −x − ∑

σ∈Γ yσ.

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Kashiwara-Vergne bigraded Lie algebra lkrv(Γ)••

• A mould M ∈ M(F; Γ) is called finite when Mm(x1, . . . , xm) = 0 except

for finitely many m. It is called polynomial-valued when Mm

σ1,...,σm(x1, . . . , xm) ∈ Q[x1, . . . , xm] for all (σ1, . . . , σm) ∈ Γ⊕m and m.

• We denote M(F; Γ)fin,pol (resp. ARI(Γ)fin,pol) to be the subset of all

finite polynomial-valued moulds in M(F; Γ) (resp. ARI(Γ)). We note that ARI(Γ)fin,pol forms a Lie subalgebra of ARI(Γ).

Theorem ([F-K, Theorem 2.22])

We have an isomorphism of bigraded Q-linear spaces lkrv(Γ)•• ≃ ARI(Γ)push/pusnu ∩ ARI(Γ)fin,pol

al

.

Theorem ([F-K, Theorem 2.23])

The space lkrv(Γ)•• forms a bigraded Lie algebra.

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On the graded case

• Define a map teru : ARI(Γ) → ARI(Γ) by

teru(M)m( x1,

. . . , xm σ1, . . . , σm

) := Mm( x1,

. . . , xm σ1, . . . , σm

) + 1 um { Mm−1( x1,

. . . , xm−2, xm−1 + xm σ1, . . . , σm−2, σm−1

) − Mm−1( x1,

. . . , xm−1 σ1, . . . , σm−1

)} .

for M ∈ ARI(Γ).

• In [Ecalle, 2011], the following relation is introduced:

teru(M) = push ◦ mantar ◦ teru ◦ mantar(M), which is called the senary relation. Here, the map mantar : ARI(Γ) → ARI(Γ) is given by mantar(M)m(

x1, . . . , xm σ1, . . . , σm

) := (−1)m−1Mm(

xm, . . . , x1 σm, . . . , σ1

) .

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On the graded case

Definition ([F-K, Definition 2.14])

We define the Q-linear space ARI(Γ)sena/pusnu to be the subset of moulds M in ARI(Γ) which satisfy the senary relation and whose swap satisfy the pus-neutrality. It is easy to know that mantar ◦ mantar = id and grD(teru) = id. Hence, we get an embedding of Q-linear space grDARI(Γ)sena/pusnu ֒ → ARI(Γ)push/pusnu.

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On the graded case

Definition ([F-K, Definition 2.1])

Define krv(Γ)• = ⊕w≥2krv(Γ)w to be the graded Q-linear space, where krv(Γ)w is the Q-linear space consisting of Lie elements F ∈ Lw which satisfies the following relations: [x, G] + ∑

τ∈Γ

[yτ, τ(F)] = 0, (KV1) tr ◦ q ◦ πY (F(z; (yσ))) = 0 (KV2) for a certain G = G(F) ∈ Lw.

Remark (conditions of lkrv(Γ)••)

[x, G] + ∑

τ∈Γ

[yτ, τ(F)] ≡ 0 mod Fild+2

D

Lw+1, (LKV 1) tr ◦ q ◦ πY (F(z; (yσ))) ≡ 0 mod tr(Fild+1

D

Aw). (LKV 2)

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On the graded case

Remark

We see that the above krv(Γ)• with Γ = {e} recovers the original Kashiwara-Vergne Lie algebra denoted by krv0

2 introduced in [A-T] and the

depth¿1-part of krv introduced in [Raphael-Schneps, Definition 3].

Theorem ([Theorem 2.15])

We have an isomorphism of filtered Q-linear spaces krv(Γ)• ≃ ARI(Γ)sena/pusnu ∩ ARI(Γ)fin,pol

al

.

Remark ([F-K, Theorem 2.22])

There is an isomorphism of bigraded Q-linear spaces lkrv(Γ)•• ≃ ARI(Γ)push/pusnu ∩ ARI(Γ)fin,pol

al

.

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Future work

1 It is also not clear if krv(Γ)• forms a Lie algebra or not.

Is krv(Γ)• a Lie algebra?

2 The existence of an embedding

Fil2

DARI(Γ)al/il ֒

→ ARI(Γ)teru/pusnu. (cf. In our paper, we proved that there is an embedding Fil2

DARI(Γ)al/al ֒

→ ARI(Γ)push/pusnu).

3 In [A-T], Lie algebras krv0

n (n ≥ 1) are introduced, and in

[Alekseev-Kawazumi-Kuno-Naef], Lie algebras krvn (n ≥ 2) are also

• introduced. We do not know how their Lie algebras krv0

n and krvn are

related to our krv(Γ)•.

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Thank you for listening!!

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