Kashiwara-Vergne and dihedral bigraded Lie algebras in mould theory
KKASHIWARA-VERGNE AND DIHEDRALBIGRADED LIE ALGEBRAS IN MOULD THEORY
HIDEKAZU FURUSHO AND NAO KOMIYAMA
Abstract.
We introduce the Kashiwara-Vergne bigraded Lie algebra associ-ated with a finite abelian group and give its mould theoretic reformulation.By using the mould theory, we show that it includes the Goncharov’s dihedralLie algebra, which generalizes the result of Raphael and Schneps.
Contents
0. Introduction 11. Preparation on mould theory 31.1. Moulds and alternality 31.2. Flexions and ari-bracket 61.3. Swap and bialternality 81.4. Push-invariance and pus-neutrality 102. Kashiwara-Vergne Lie algebra 182.1. Γ -variant of the KV condition 182.2. Kashiwara-Vergne bigraded Lie algebra 263. Dihedral Lie algebra 293.1. Dihedral bigraded Lie algebra 293.2. Mould theoretic reformulation 313.3. Embedding 33Appendix A. On the ari -bracket of ARI(Γ)
Introduction
The dihedral Lie algebra D (Γ) •• is the bigraded Lie algebra introduced in [G01a]which is associated with a finite abelian group Γ . It reflects the double shuffle anddistribution relations among multiple polylogarithms evaluated at roots of unity. Itsrelation with a certain bigraded variant of motivic Lie algebra is discussed in loc. cit. Date : June 1, 2020. a r X i v : . [ m a t h . QA ] J un HIDEKAZU FURUSHO AND NAO KOMIYAMA
Kashiwara-Vergne Lie algebra krv • is the filtered graded Lie algebra introducedin [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 tobe isomorphic to the motivic Lie algebra (cf. [F]).A bigraded variant lkrv •• of krv • is introduced and its mould theoretical inter-pretation [Ec81] is investigated in [RS]. The results in [M, RS] give an inclusion ofbigraded Lie algebras(0.1) D ( { e } ) •• (cid:44) → lkrv •• . Our objective of this paper is to extend it to any Γ by exploiting Ecalle’s mouldtheory ([Ec03, Ec11]) with self-contained proofs. Our results are exhibited as fol-lows:(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 identifiedwith the Q -linear space of finite polynomial-valued alternal moulds satisfy-ing Ecalle’s senary relation (2.14) and whose swap ’s (1.7) are pus -neutral(1.8), that is, there is an isomorphism of Q -linear spaces krv (Γ) • (cid:39) ARI(Γ) sena / pusnu ∩ ARI(Γ) fin , polal . (ii) In Definition 2.17, we introduce a bigraded Q -linear space lkrv (Γ) •• whichis defined by the ‘leading terms’ of the defining equations of krv (Γ) • . Wealso consider its subspace lkrvd (Γ) •• by imposing the distribution relation inDefinition 2.25. Both of them recover lkrv •• when Γ = { e } . It is shown thatthey form Lie algebras in Theorem 2.23 and Corollary 2.26. An inclusionof bigraded Lie algebras gr D krv (Γ) • (cid:44) → lkrv (Γ) •• is presented in (2.31), where the first term means the associated bigradedof the filtered graded linear space krv (Γ) • .(iii) In Theorem 2.22, we show that lkrv (Γ) •• is identified with the Lie algebra(cf. Theorem 1.28) of finite polynomial-valued alternal moulds which are push -invariant (1.4) and whose swap ’s are pus -neutral (1.8), that is, thereis an isomorphism of Lie algebras lkrv (Γ) •• (cid:39) ARI(Γ) push / pusnu ∩ ARI(Γ) fin , polal . (iv) In §3, we consider Goncharov’s dihedral bigraded Lie algebra D (Γ) •• andits related Lie algebra D (Γ) •• with the dihedral symmetry which contains D (Γ) •• . It is explained in Theorem 3.5 that its depth>1-part of D (Γ) •• coincides with the depth>1-part of the Lie algebra of finite polynomial-valued part of the set of moulds ARI(Γ) al / al (cf. Definition 1.20), namely Fil D D (Γ) •• (cid:39) Fil D ARI(Γ) fin , polal / al . In Theorem 3.13, we show that there is an inclusion of graded Lie algebras
Fil D ARI(Γ) al / al (cid:44) → ARI(Γ) push / pusnu . In Corollary 3.14, by taking an intersection with
ARI(Γ) fin , polal we obtainthe inclusion of bigraded Lie algebras from the depth >1 part Fil D D (Γ) •• of D (Γ) •• to lkrv (Γ) •• Fil D D (Γ) •• (cid:44) → lkrv (Γ) •• , ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 3 which extends (0.1). By imposing the distribution relation there the inclu-sion
Fil D D (Γ) •• (cid:44) → lkrvd (Γ) •• is similarly obtained in Corollary 3.15.In Appendix A, we give self-contained proofs of several fundamental propertieson the ari -bracket of the Lie algebra ARI(Γ) of moulds associated with Γ . InAppendix B, we discuss moulds arising from the multiple polylogarithms evaluatedat roots of unity. Acknowledgements.
We thank for L. Schneps who gave comments on the first ver-sion of the paper and informing us [RS]. H.F. and N.K. have been supported bygrants JSPS KAKENHI JP18H01110 and JP18J14774 respectively.1.
Preparation on mould theory
We prepare several techniques of moulds which will be employed in our latersections. The notion of moulds, the alternality, flexions and the ari-bracket associ-ated with a finite group Γ are explained in §1.1 and §1.2. In §1.3, we explain thatthe 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 formsa Lie algebra under the ari -bracket in Theorem 1.28.1.1. Moulds and alternality.
We introduce and discuss moulds associated witha finite abelian group Γ .The notion of moulds was invented by Ecalle (cf. [Ec81, Tome I. pp.12-13]). Forour convenience we employ the following formulation influenced by [Sch15] whichis different from the one employed in [C, Définition 1 or Définition II.1] and [Sau,§4.1].Let Γ be a finite abelian group. We set F := (cid:83) m (cid:62) Q ( x , . . . , x m ) . Definition 1.1. A mould on Z (cid:62) with values in F is a collection (a sequence) M = ( M m ( x , . . . , x m )) m ∈ Z (cid:62) = (cid:0) M ( ∅ ) , M ( x ) , M ( x , x ) , . . . (cid:1) , with M ( ∅ ) ∈ Q and M m ( x , . . . , x m ) ∈ Q ( x , . . . , x m ) ⊕ Γ ⊕ m for m (cid:62) , which isdescribed by a summation M m ( x , . . . , x m ) = ⊕ ( σ ,...,σ m ) ∈ Γ ⊕ m M mσ ,...,σ m ( x , . . . , x m ) where each M mσ ,...,σ m ( x , . . . , x m ) ∈ Q ( x , . . . , x m ) . We denote the set of allmoulds with values in F (Γ) by M ( F ; Γ) . The set M ( F ; Γ) forms a Q -linear spaceby A + B := ( A m ( x , . . . , x m ) + B m ( x , . . . , x m )) m ∈ Z (cid:62) ,cA := ( cA m ( x , . . . , x m )) m ∈ Z (cid:62) , for A, B ∈ M ( F ; Γ) and c ∈ Q , namely the addition and the scalar are takencomponentwise. We define a product on M ( F ; Γ) by ( A × B ) mσ ,...,σ m ( x , . . . , x m ) := m (cid:88) i =0 A iσ ,...,σ i ( x , . . . , x i ) B m − iσ i +1 ,...,σ m ( x i +1 , . . . , x m ) , for A, B ∈ M ( F ; Γ) and for m (cid:62) and for ( σ , . . . , σ m ) ∈ 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 , , , . . . ) . HIDEKAZU FURUSHO AND NAO KOMIYAMA
By the regular action of Γ on Q [Γ] , M ( F ; Γ) admits the action of Γ . It isdescribed by ( γM ) mσ ,...,σ m ( x , . . . , x m ) = M mγ − σ ,...,γ − σ m ( x , . . . , x m ) for γ ∈ Γ .The set M ( F : Γ) is encoded with the depth filtration { Fil m D M ( F ; Γ) } m (cid:62) where Fil m D M ( F ; Γ) is the collection of moulds with M r ( x , . . . , x r ) = 0 for r < m . It isclear that the algebra structure of M ( F ; Γ) is compatible with the depth filtration.Put ARI(Γ) := { M ∈ M ( F ; Γ) | M ( ∅ ) = 0 } . It is a filtered (non-unital) subalgebra.
Remark 1.2.
Ecalle introduced the notion of bimoulds in [Ec03, §5] and [Ec11,§1.3] (cf. [Sch15, §2.1]) which are the so-called ‘moulds with double layered indices’.We often regard our moulds in Definition 1.1 as if a bimould and we sometimesdenote M mσ ,...,σ m ( x , . . . , x m ) by M m (cid:0) x , ..., x m σ , ..., σ m (cid:1) for M ∈ ARI(Γ) as in loc. cit.
Definition 1.3.
A mould M ∈ M ( F ; Γ) is called finite when M m ( x , . . . , x m ) = 0 except for finitely many m . It is called polynomial-valued when M mσ ,...,σ m ( x , . . . , x m ) ∈ Q [ x , . . . , x m ] for all ( σ , . . . , σ m ) ∈ 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 prepare the following algebraic formulation which is useful to present thenotion of the alternality of mould: Put X := (cid:8)(cid:0) x i σ (cid:1)(cid:9) i ∈ N ,σ ∈ Γ . Let X Z be the set suchthat X Z := { ( uσ ) | u = a x + · · · + 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 · · · u m ∈ X • Z with u , . . . , u m ∈ X Z by ω = ( u , . . . , u m ) as a sequence. The length of ω = u · · · u m is defined to be l ( ω ) := m .For our simplicity we occasionally denote M ∈ M ( F ; Γ) by M = ( M m ( x m )) m ∈ Z (cid:62) or M = ( M ( x m )) m ∈ Z (cid:62) , where x := ∅ and x m := (cid:0) x , ..., x m σ , ..., σ m (cid:1) for m (cid:62) . Under the notations, the productof A, B ∈ ARI(Γ) is expressed as A × B = Ñ (cid:88) x m = αβ A l ( α ) ( α ) B l ( β ) ( β ) é m ∈ Z (cid:62) where α and β run over X • Z .We set A X := Q (cid:104) X Z (cid:105) to be the non-commutative polynomial algebra generatedby X Z (i.e. A X is the Q -linear space generated by X • Z ). We equip A X a product X : A ⊗ X → A X 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, asso-ciative, unital Q -algebra. We should beware of the inequality (cid:0) x + x σ (cid:1) (cid:54) = (cid:0) x σ (cid:1) + (cid:0) x σ (cid:1) and (cid:0) σ (cid:1) (cid:54) = 0 . ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 5
We define the family (cid:8) Sh (cid:0) ω ; ηα (cid:1)(cid:9) ω,η,α ∈ X • Z in Z by ω X η = (cid:88) α ∈ X • Z Sh Ç ω ; ηα å α. Particularly for p, q ∈ N and u , . . . , u p + q ∈ X Z , we rewrite the shuffle product by ( u , . . . , u p ) X ( u p +1 , . . . , u p + q ) = (cid:88) σ ∈ X p,q ( u σ (1) , . . . , u σ ( p ) , u σ ( p +1) , . . . , u σ ( p + q ) ) . Here the set X p,q is defined by(1.2) { σ ∈ S p + q | σ − (1) < · · · < σ − ( p ) , σ − ( p + 1) < · · · < σ − ( p + q ) } , where S p + q is the symmetry group with degree p + q . Definition 1.4.
A mould M ∈ ARI(Γ) is called alternal (cf. [Ec81, I–p.118]) if(1.3) (cid:88) α ∈ X • Z Sh Ç (cid:0) x , ..., x p σ , ..., σ p (cid:1) ; (cid:0) x p +1 , ..., x p + q σ p +1 , ..., σ p + q (cid:1) α å M p + q ( α ) = 0 , for all p, q (cid:62) . The Q -linear space ARI(Γ) al is defined to be the subset of moulds M ∈ ARI(Γ) which are alternal (cf. [Ec03], [Ec11]).We encode it with the induced depth filtrations and it forms a filtered Q -algebra. Remark 1.5.
Assume that u , . . . , u m ∈ F are algebraically independent over Q .For M ∈ ARI(Γ) we denote M m (cid:0) u , ..., u m σ , ..., σ m (cid:1) to be the image of M m (cid:0) x , ..., x m σ , ..., σ m (cid:1) under the field embedding Q ( x , . . . , x m ) (cid:44) → F sending x i (cid:55)→ u i .For our later use, we prepare more notations: Notation 1.6 ([Ec11, §2.1]) . For any mould M = (cid:0) M m (cid:0) u , ..., u m σ , ..., σ m (cid:1)(cid:1) m ∈ ARI(Γ) ,we define mantar( M ) m Ä u , . . . , u m σ , . . . , σ m ä = ( − m − M m Ä u m , . . . , u σ m , . . . , σ ä , push( M ) m Ä u , . . . , u m σ , . . . , σ m ä = M m Ä − u − · · · − u m , u , . . . , u m − σ − m , σ σ − m , . . . , σ m − σ − m ä , neg( M ) m Ä u , . . . , u m σ , . . . , σ m ä = M m (cid:16) − u , . . . , − u m σ − , . . . , σ − m (cid:17) , teru( M ) m Ä u , . . . , u m σ , . . . , σ m ä = M m Ä u , . . . , u m σ , . . . , σ m ä + 1 u m ¶ M m − Ä u , . . . , u m − , u m − + u m σ , . . . , σ m − , σ m − ä − M m − Ä u , . . . , u m − σ , . . . , σ m − ä© . Note that they are all Q -linear endomorphisms on ARI(Γ) . We remark that neg ◦ neg = id and mantar ◦ mantar = id . Definition 1.7.
We call a mould M ∈ ARI(Γ) push-invariant when we have(1.4) push( M ) = M. We define
ARI(Γ) push ([Ec11, §2.5]) to be the set of moulds M in ARI(Γ) which ispush-invariant (1.4).
HIDEKAZU FURUSHO AND NAO KOMIYAMA
Flexions and ari-bracket.
We explain an ari-bracket ari u on ARI(Γ) byusing flexions.The notion of flexions is introduced by Ecalle in [Ec11, §2.1] for bimoulds (cf.[Sch15, §2.2]). Here we consider those for moulds in
ARI(Γ) . Definition 1.8.
The flexions are the four binary operators ∗ (cid:100)∗ , ∗(cid:101) ∗ , ∗ (cid:98)∗ , ∗(cid:99) ∗ : X • Z × X • Z → X • Z which are defined by β (cid:100) α := Ä b + · · · + b n + a , a , . . . , a m σ , σ , . . . , σ m ä ,α (cid:101) β := Ä a , . . . , a m − , a m + b + · · · + b n σ , . . . , σ m − , σ m ä , β (cid:98) α := Ä a , . . . , a m τ − n σ , . . . , τ − n σ m ä ,α (cid:99) β := (cid:16) a , . . . , a m σ τ − , . . . , σ m τ − (cid:17) , ∅ (cid:100) γ := γ (cid:101) ∅ := ∅ (cid:98) γ := γ (cid:99) ∅ := γ, γ (cid:100)∅ := ∅(cid:101) γ := γ (cid:98)∅ := ∅(cid:99) γ := ∅ , for α = (cid:0) a ,...,a m σ ,...,σ m (cid:1) , β = (cid:0) b ,...,b n τ ,...,τ n (cid:1) ∈ X • Z ( m, n (cid:62) ) and γ ∈ X • Z .Note that we have l ( β (cid:100) α ) = l ( α (cid:101) β ) = l ( β (cid:98) α ) = l ( α (cid:99) β ) = l ( α ) and l ( α, β ) = l ( α ) + l ( β ) for α, β ∈ X • Z .The derivation arit and bracket ari are introduced for bimoulds in terms offlexions in [Ec11, §2.2] (cf. [Sch15, §2.2]) and here we consider those for ARI(Γ) asfollows.
Definition 1.9.
Let B ∈ ARI(Γ) . The linear map arit u ( B ) : ARI(Γ) → ARI(Γ) is defined by (arit u ( B )( A )) m ( x m ) = (arit u ( B )( A )) m (cid:0) x , ..., x m σ , ..., σ m (cid:1) := (cid:88) x m = αβγβ,γ (cid:54) = ∅ A l ( α,γ ) ( α β (cid:100) γ ) B l ( β ) ( β (cid:99) γ ) − (cid:88) x m = αβγα,β (cid:54) = ∅ A l ( α,γ ) ( α (cid:101) β γ ) B l ( β ) ( α (cid:98) β ) ( m (cid:62) , m = 0 , , for A ∈ ARI(Γ) .It is shown in [Sch15, Appendix A.1] that arit forms a derivation on the set ofbimoulds. The same holds for
ARI(Γ) as follows, where we present an alternativeproof to hers.
Lemma 1.10.
For any A ∈ ARI(Γ) , arit u ( A ) forms a derivation of ARI(Γ) withrespect to the product × , that is, for any B, C ∈ ARI(Γ) , we have arit u ( A )( B × C ) = arit u ( A )( B ) × C + B × arit u ( A )( C ) . The lower suffix u is reflected by the notion of u -moulds in [Sch15]. For m (cid:62) and α, β, γ ∈ X • Z with x m = αβγ , all letters appearing in the word α β (cid:100) γ (resp. β (cid:98) γ ) are algebraically independent over Q . So by Remark 1.5, the component A l ( α,γ ) ( α β (cid:100) γ ) (resp. B l ( β ) ( β (cid:99) γ ) ) is well-defined. Similarly, A l ( α,γ ) ( α (cid:101) β γ ) and B l ( β ) ( α (cid:98) β ) are also well-defined. ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 7
Proof.
Let m (cid:62) . We have (arit( A )( B × C ))( x m ) = (cid:88) x m = abcb,c (cid:54) = ∅ ( B × C )( a b (cid:100) c ) A ( b (cid:99) c ) − (cid:88) x m = abca,b (cid:54) = ∅ ( B × C )( a (cid:101) b c ) A ( a (cid:98) b )= (cid:88) x m = abcb,c (cid:54) = ∅ (cid:40) (cid:88) a b (cid:100) c = de B ( d ) C ( e ) (cid:41) A ( b (cid:99) c ) − (cid:88) x m = abca,b (cid:54) = ∅ (cid:40) (cid:88) a (cid:101) b c = ef B ( e ) C ( f ) (cid:41) A ( a (cid:98) b ) By applying Lemma A.2.(1) for c (cid:54) = ∅ , f = ∅ to the first term and by applyingLemma A.2.(2) for a (cid:54) = ∅ , d = ∅ to the second term, we have = (cid:88) x m = abcb,c (cid:54) = ∅ (cid:88) a = a a B ( a ) C ( a b (cid:100) c ) + (cid:88) c = c c c (cid:54) = ∅ B ( a b (cid:100) c ) C ( c ) A ( b (cid:99) c ) − (cid:88) x m = abca,b (cid:54) = ∅ (cid:88) a = a a a (cid:54) = ∅ B ( a ) C ( a (cid:101) b c ) + (cid:88) c = c c B ( a (cid:101) b c ) C ( c ) A ( a (cid:98) b )= (cid:88) x m = a a bcb,c (cid:54) = ∅ B ( a ) C ( a b (cid:100) c ) A ( b (cid:99) c ) + (cid:88) x m = abc c b,c (cid:54) = ∅ B ( a b (cid:100) c ) C ( c ) A ( b (cid:99) c c ) − (cid:88) x m = a a bca ,b (cid:54) = ∅ B ( a ) C ( a (cid:101) b c ) A ( a a (cid:98) b ) − (cid:88) x m = abc c a,b (cid:54) = ∅ B ( a (cid:101) b c ) C ( c ) A ( a (cid:98) b ) . By applying Lemma A.1 to the second term, we get A ( b (cid:99) c c ) = A ( b (cid:99) c ) . Similarly,for the third term, we get A ( a a (cid:98) b ) = A ( a (cid:98) b ) . Therefore, we calculate = (cid:88) x m = dc (cid:88) d = abc b,c (cid:54) = ∅ B ( a b (cid:100) c ) A ( b (cid:99) c ) − (cid:88) d = abc a,b (cid:54) = ∅ B ( a (cid:101) b c ) A ( a (cid:98) b ) C ( c )+ (cid:88) x m = a d B ( a ) (cid:88) d = a bcb,c (cid:54) = ∅ C ( a b (cid:100) c ) A ( b (cid:99) c ) − (cid:88) d = a bca ,b (cid:54) = ∅ C ( a (cid:101) b c ) A ( a (cid:98) b ) = (cid:88) x m = dc (arit( A )( B ))( d ) C ( c ) + (cid:88) w n = a d B ( a )(arit( A )( C ))( d )=(arit( A )( B ) × C )( x m ) + ( B × arit( A )( C ))( x m ) . Hence, we obtain the claim. (cid:3)
We define the following bracket as with the bracket ari introduced in [Ec11,(2.40)].
Definition 1.11.
The ari u -bracket means the bilinear map ari u : ARI(Γ) ⊗ → ARI(Γ) which is defined by(1.5) ari u ( A, B ) := arit u ( B )( A ) − arit u ( A )( B ) + [ A, B ] for A, B ∈ ARI(Γ) . Here , we have [ A, B ] := A × B − B × A . In the papers [Ec11], [Sch15] and [RS], the product A × B (resp. the bracket [ A, B ] ) is denotedby mu ( A, B ) (resp. lu ( A, B ) ). HIDEKAZU FURUSHO AND NAO KOMIYAMA
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 1.12.
The Q -linear space ARI(Γ) forms a filtered Lie algebra underthe ari u -bracket.Proof. We give a self-contained proof in Appendix A.1. (cid:3)
The following proposition for
Γ = { e } is shown in [SaSch, Appendix A]. Proposition 1.13.
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. (cid:3)
Swap and bialternality.
We encode
ARI(Γ) with another Lie algebra struc-ture introduced by ari v . We prepare ARI(Γ) , a copy of
ARI(Γ) . We denote M mσ ,...,σ m ( x , . . . , x m ) by M m (cid:0) σ , ..., σ m x , ..., x m (cid:1) for each element M in ARI(Γ) todistinguish it from an element in
ARI(Γ) .Similarly to our previous sections, we work over the following algebraic formu-lation: Put Y := ¶ (cid:0) σy i (cid:1) © i ∈ N ,σ ∈ Γ . Let Y Z be the set such that Y Z := { ( σv ) | v = a y + · · · + a k y k , k ∈ N , a j ∈ Z , σ ∈ Γ } , 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 (cid:104) Y Z (cid:105) to be the non-commutativepolynomial algebra generated by Y Z . In the same way to A X , it is equipped with astructure of a commutative, associative, unital Q -algebra with the shuffle product X : A ⊗ Y → A Y . The flexions are also introduced in this setting. Definition 1.14.
The flexions on Y • Z are the four binary operators ∗ (cid:100)∗ , ∗(cid:101) ∗ , ∗ (cid:98)∗ , ∗(cid:99) ∗ : Y • Z × Y • Z → Y • Z which are defined by β (cid:100) α := Ä σ τ · · · τ n , σ , . . . , σ m a , a , . . . , a m ä ,α (cid:101) β := Ä σ , . . . , σ m − , σ m τ · · · τ n a , . . . , a m − , a m ä , β (cid:98) α := Ä σ , . . . , σ m a − b n , . . . , a m − b n ä ,α (cid:99) β := Ä σ , . . . , σ m a − b , . . . , a m − b ä , ∅ (cid:100) γ := γ (cid:101) ∅ := ∅ (cid:98) γ := γ (cid:99) ∅ := γ, γ (cid:100)∅ := ∅(cid:101) γ := γ (cid:98)∅ := ∅(cid:99) γ := ∅ , for α = (cid:0) σ ,...,σ m a ,...,a m (cid:1) , β = Ä τ ,...,τ n b ,...,b n ä ∈ Y • Z ( m, n (cid:62) ) and γ ∈ Y • Z .We denote y := ∅ and y m := (cid:0) σ , ..., σ m y , ..., y m (cid:1) for m (cid:62) . We note that the top and bottom rows are switched.
ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 9
Definition 1.15.
Let B ∈ ARI(Γ) . The linear map arit v ( B ) : ARI(Γ) → ARI(Γ) is defined by (arit v ( B )( A )) m ( y m ) = (arit v ( B )( A )) m (cid:0) σ , ..., σ m y , ..., y m (cid:1) := (cid:88) y m = αβγβ,γ (cid:54) = ∅ A l ( α,γ ) ( α β (cid:100) γ ) B l ( β ) ( β (cid:99) γ ) − (cid:88) y m = αβγα,β (cid:54) = ∅ A l ( α,γ ) ( α (cid:101) β γ ) B l ( β ) ( α (cid:98) β ) ( m (cid:62) , m = 0 , , for A ∈ ARI(Γ) .Similarly to Lemma 1.10, the following holds.
Lemma 1.16.
For any A ∈ ARI(Γ) , arit v ( A ) forms a derivation of ARI(Γ) withrespect to the product × .Proof. The arit v -bracket can be expressed by the exactly same formula as the arit u -bracket in terms of flexions. Therefore it can be proved in the same way to the oneof Lemma 1.10. (cid:3) Definition 1.17.
The ari v -bracket means the bilinear map ari v : ARI(Γ) ⊗ → ARI(Γ) which is defined by (1.6) ari v ( A, B ) := arit v ( B )( A ) − arit v ( A )( B ) + [ A, B ] for A, B ∈ ARI(Γ) . Here [ A, B ] := A × B − B × A .Similarly to Proposition 1.12, the following holds. Proposition 1.18.
The Q -linear space ARI(Γ) forms a filtered Lie algebra underthe ari v -bracket.Proof. It can be also proved in the same way to the one of Proposition 1.12. (cid:3)
In the same way to Definition 1.4, alternal moulds in
ARI(Γ) can be introducedand we denote
ARI(Γ) al to be its subset consisting of alternal moulds. Similarly toProposition 1.13, the following holds. Proposition 1.19.
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 1.13. (cid:3)
We define the Q -linear map swap : ARI(Γ) → ARI(Γ) by(1.7) swap( M ) m Ä σ , . . . , σ m v , . . . , v m ä = M m Ä v m , v m − − v m , . . . , v − v , v − v σ · · · σ m , σ · · · σ m − , . . . , σ σ , σ ä for any mould M = (cid:0) M m (cid:0) u , ..., u m σ , ..., σ m (cid:1)(cid:1) ∈ ARI(Γ) . Definition 1.20 (cf. [Ec03], [Ec11]) . The subset
ARI(Γ) al / al of bialternal mouldsis defined to be ARI(Γ) al / al := { M ∈ ARI(Γ) al | swap( M ) ∈ ARI(Γ) al , M (cid:0) x σ (cid:1) = M (cid:16) − x σ − (cid:17) } . Proposition 1.21.
The Q -linear space ARI(Γ) al / al forms a filtered Lie subalgebraof ARI(Γ) al under the ari u -bracket.Proof. We prove this in Appendix A.3. (cid:3) The lower suffix v is reflected by the notion of v -moulds in [Sch15]. In [RS], the brackets arit v ( A, B ) and ari v ( A, B ) are denoted by arit and ari respectively. Push-invariance and pus-neutrality.
We introduce the set
ARI(Γ) push / pusnu of moulds which are push-invariant and whose swap are pus-neutral and show thatit forms a Lie algebra under the ari-bracket.By abuse of notation, we introduce with the following notations for ARI(Γ) similarly to Notation 1.6.
Notation 1.22 ([Ec11, §2.1]) . For any mould M = (cid:0) M m (cid:0) σ , ..., σ m v , ..., v m (cid:1)(cid:1) m ∈ ARI(Γ) ,we define mantar( M ) m Ä σ , . . . , σ m v , . . . , v m ä = ( − m − M m Ä σ m , . . . , σ v m , . . . , v ä , pus( M ) m Ä σ , . . . , σ m v , . . . , v m ä = M m Ä σ m , σ , . . . , σ m − v m , v , . . . , v m − ä , neg( M ) m Ä σ , . . . , σ m v , . . . , v m ä = M m (cid:16) σ − , . . . , σ − m − u , . . . , − u m (cid:17) . Definition 1.23.
We call a mould N ∈ ARI(Γ) pus-neutral ([Ec11, (2.73)]) whenwe have(1.8) m (cid:88) i =1 pus i ( N ) m (cid:0) σ , ..., σ m v , ..., v m (cid:1) Ñ = (cid:88) i ∈ Z /m Z N m Ä σ i +1 , ..., σ i + m v i +1 , ..., v i + m äé = 0 for all m (cid:62) and σ , . . . , σ m ∈ Γ . We define ARI(Γ) push / pusnu to be the setof moulds M ∈ ARI(Γ) which is push-invariant (1.4) and whose swap( M ) is pus-neutral (1.8).Firstly we show that the set ARI push forms a Lie algebra under the ari u -bracketwhich was stated in [Ec11, §2.5] without a detailed proof. Proposition 1.24. If A, B ∈ ARI(Γ) are push -invariant, then ari u ( A, B ) is push -invariant.Proof. Let m (cid:62) . We put ω = (cid:0) u , ..., u m σ , ..., σ m (cid:1) . We have push( A × B )( ω )=( A × B ) Ä − u − · · · − u m , u , . . . , u m − σ − m , σ σ − m , . . . , σ m − σ − m ä = A ( ∅ ) B Ä − u − · · · − u m , u , . . . , u m − σ − m , σ σ − m , . . . , σ m − σ − m ä + m (cid:88) i =1 A Ä − u − · · · − u m , u , . . . , u i − σ − m , σ σ − m , . . . , σ i − σ − m ä B Ä u i , . . . , u m − σ i σ − m , . . . , σ m − σ − m ä . Because
A, B are push -invariant, we get push( A × B )( ω ) (1.9) = A ( ∅ ) B ( ω ) + A ( ω ) B ( ∅ )+ m − (cid:88) i =1 A Ä u , . . . , u i − , u i + · · · + u m σ , . . . , σ i − , σ m ä B Ä u i , . . . , u m − σ i σ − m , . . . , σ m − σ − m ä . It is not push-neutral but pus-neutral. In [RS, Definition 5’], it is called circ-neutral when(1.8) holds for m > when Γ = { e } . We expect that our
ARI(Γ) push / pusnu might be related to Ecalle’s ARI ∗ / pusnu ([Ec11, §2.5]),whose precise definition looks missing. ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 11
On the other hand, for η = (cid:0) u , u , ..., u m − τ , τ , ..., τ m − (cid:1) , we have arit u ( B )( A )( η )= (cid:88) η = αβγβ,γ (cid:54) = ∅ A ( α β (cid:100) γ ) B ( β (cid:99) γ ) − (cid:88) η = αβγα,β (cid:54) = ∅ A ( α (cid:101) β γ ) B ( α (cid:98) β )= (cid:88) (cid:54) i (cid:54) j A, B are push -invariant, we get = (cid:88) (cid:54) j This proposition improves the results of [RS, §4.1.3], which showsthat the intersection ARI push (Γ) ∩ ARI polal (Γ) forms a Lie algebra under the ari u -bracket when Γ = { e } . ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 13 Secondly we show that the set of pus-neutral moulds is closed under the ari v -bracket. The following proposition is a generalization of [RS, Lemma 21] whichtreats the case when Γ = { e } . Proposition 1.26. If A, B ∈ ARI(Γ) are pus -neutral, then ari v ( A, B ) is pus -neutral.Proof. The proof goes in the same way to that of [RS]. Let m (cid:62) . Since thealgebra ARI(Γ) is graded by depth, it is enough to prove A and B ∈ ARI(Γ) withdepth k and l ∈ N with m = k + l .Firstly, we prove that [ A, B ] is pus -neutral. We calculate m (cid:88) i =1 pus i ([ A, B ]) m (cid:0) σ , ..., σ m v , ..., v m (cid:1) = (cid:88) i ∈ Z /m Z ( A × B − B × A ) m Ä σ i , ..., σ m + i v i , ..., v m + i ä = (cid:88) i ∈ Z /m Z ¶ A k Ä σ i , ..., σ k + i v i , ..., v k + i ä B l Ä σ k +1+ i , ..., σ m + i v k +1+ i , ..., v m + i ä − B l Ä σ i , ..., σ l + i v i , ..., v l + i ä A k Ä σ l +1+ i , ..., σ m + i v l +1+ i , ..., v m + i ä© = (cid:88) i ∈ Z /m Z ¶ A k Ä σ i , ..., σ k + i v i , ..., v k + i ä B l Ä σ k +1+ i , ..., σ m + i v k +1+ i , ..., v m + i ä − B l Ä σ i − l , ..., σ i v i − l , ..., v i ä A k Ä σ i , ..., σ k + i v i , ..., v k + i ä© = (cid:88) i ∈ Z /m Z ¶ A k Ä σ i , ..., σ k + i v i , ..., v k + i ä B l Ä σ k +1+ i , ..., σ m + i v k +1+ i , ..., v m + i ä − B l Ä σ k +1+ i , ..., σ m + i v k +1+ i , ..., v m + i ä A k Ä σ i , ..., σ k + i v i , ..., v k + i ä© The first and the second terms are equal, so [ A, B ] is pus -neutral.Secondly, we prove that arit v ( B )( A ) is pus -neutral. Then we have (arit v ( B )( A )) m (cid:0) σ , ..., σ m v , ..., v m (cid:1) = (cid:88) ω = αβγβ,γ (cid:54) = ∅ A k ( α β (cid:100) γ ) B l ( β (cid:99) γ ) − (cid:88) ω = αβγα,β (cid:54) = ∅ A k ( α (cid:101) β γ ) B l ( α (cid:98) β ) . Because l (cid:62) and B is with depth l , we may put α = (cid:0) σ , ..., σ j v , ..., v j (cid:1) , β = (cid:0) σ j +1 , ..., σ j + l v j +1 , ..., v j + l (cid:1) and γ = (cid:0) σ j + l +1 , ..., σ m v j + l +1 , ..., v m (cid:1) and we get (arit v ( B )( A )) m (cid:0) σ , ..., σ m v , ..., v m (cid:1) = k − (cid:88) j =0 A k Ä σ , . . . , σ j , σ j +1 · · · σ j + l +1 , σ j + l +2 , . . . , σ m v , . . . , v j , v j + l +1 , v j + l +2 , . . . , v m ä B l Ä σ j +1 , . . . , σ j + l v j +1 − v j + l +1 , . . . , v j + l − v j + l +1 ä − k (cid:88) j =1 A k Ä σ , . . . , σ j − , σ j · · · σ j + l , σ j + l +1 , . . . , σ m v , . . . , v j − , v j , v j + l +1 , . . . , v m ä B l Ä σ j +1 , . . . , σ j + l v j +1 − v j , . . . , v j + l − v j ä . For the first (resp. second) term, the word α (resp. γ ) can be ∅ and the word γ (resp. α ) isnot ∅ . So the index j must run from (resp. ) to k − m − l − (resp. k = m − l ). Here, for (cid:54) j (cid:54) k − , we have (cid:88) i ∈ Z /m Z k − (cid:88) j =0 A k Ä σ i +1 , . . . , σ i + j , σ i + j +1 · · · σ i + j + l +1 , σ i + j + l +2 , . . . , σ i + m v i +1 , . . . , v i + j , v i + j + l +1 , v i + j + l +2 , . . . , v i + m ä · B l Ä σ i + j +1 , . . . , σ i + j + l v i + j +1 − v i + j + l +1 , . . . , v i + j + l − v i + j + l +1 ä = (cid:88) i ∈ Z /m Z k − (cid:88) j =0 A k Ä σ i +1 − j , . . . , σ i , σ i +1 · · · σ i + l +1 , σ i + l +2 , . . . , σ i + m − j v i +1 − j , . . . , v i , v i + l +1 , v i + l +2 , . . . , v i + m − j ä · B l Ä σ i +1 , . . . , σ i + l v i +1 − v i + l +1 , . . . , v i + l − v i + l +1 ä . = (cid:88) i ∈ Z /m Z k − (cid:88) j =0 (pus j ( A )) k Ä σ i +1 · · · σ i + l +1 , σ i + l +2 , . . . , σ i + m v i + l +1 , v i + l +2 , . . . , v i + m ä · B l Ä σ i +1 , . . . , σ i + l v i +1 − v i + l +1 , . . . , v i + l − v i + l +1 ä . Since A is pus -neutral, this is 0. Similarly by using the pus -neutrality of A , weobtain (cid:88) i ∈ Z /m Z k (cid:88) j =1 A k Ä σ i +1 , . . . , σ i + j − , σ i + j · · · σ i + j + l , σ i + j + l +1 , . . . , σ i + m v i +1 , . . . , v i + j − , v i + j , v i + j + l +1 , . . . , v i + m ä · B l Ä σ i + j +1 , . . . , σ i + j + l v i + j +1 − v i + j , . . . , v i + j + l − v i + j ä = 0 for all (cid:54) j (cid:54) k . Therefore, we get m (cid:88) i =1 pus i (arit v ( B )( A )) m (cid:0) σ , ..., σ m v , ..., v m (cid:1) = (cid:88) i ∈ Z /m Z (arit v ( B )( A )) m Ä σ i , ..., σ m + i v i , ..., v m + i ä = 0 . Whence arit v ( B )( A ) is pus -neutral.Thirdly, in the same way, we can show that arit v ( A )( B ) is pus -neutral by usingthe pus -neutrality of B .Thus we complete the proof because ari v ( A, B ) = [ A, B ]+arit v ( B )( A ) − arit v ( A )( B ) . (cid:3) We need the following lemma for the proof of Theorem 1.28. Lemma 1.27. If A, B ∈ ARI(Γ) are push -invariant, then we have swap(ari u ( A, B )) = ari v (swap( A ) , swap( B )) . Proof. This follows from the proof of [Sch15, Lemma 2.4.1], which actually worksfor BARI . (cid:3) Theorem 1.28. The set ARI(Γ) push / pusnu forms a Lie subalgebra of ARI(Γ) push under the ari u -bracket.Proof. Let A, B ∈ ARI(Γ) push / pusnu . Then by Proposition 1.24, ari u ( A, B ) ∈ ARI(Γ) push . Let m (cid:62) . Because A and B are push -invariant, by Lemma 1.27, ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 15 we have (cid:88) i ∈ Z /m Z pus i ◦ swap(ari u ( A, B )) m (cid:0) σ , ..., σ m v , ..., v m (cid:1) = (cid:88) i ∈ Z /m Z pus i (cid:0) ari v (swap( A ) , swap( B )) (cid:1) m (cid:0) σ , ..., σ m v , ..., v m (cid:1) . By Proposition 1.26, the right hand side is equal to 0. Hence swap(ari u ( A, B )) ispus-neutral. Thus we obtain ari u ( A, B ) ∈ ARI(Γ) push / pusnu . (cid:3) Remark 1.29. The above theorem improves [RS, Corollary 22] which shows that ARI(Γ) push / pusnu ∩ ARI polal (Γ) forms a Lie algebra when Γ = { e } .Our Lie algebra ARI(Γ) push / pusnu will play an important role in the followingsections. Definition 1.30. For N (cid:62) , put Γ N := { g N ∈ Γ | g ∈ Γ } . We consider the map i N : ARI(Γ) → ARI(Γ N ) which is given by(1.11) i N ( M ) m (cid:0) x , ..., x m σ , ..., σ m (cid:1) = M m (cid:0) x , ..., x m σ , ..., σ m (cid:1) and also the map m N : ARI(Γ) → ARI(Γ N ) which is given by(1.12) m N ( M ) m (cid:0) x , ..., x m σ , ..., σ m (cid:1) = (cid:88) τ Ni = σ i M m (cid:0) Nx , ..., Nx m τ , ..., τ m (cid:1) . We define the following Q -linear subspaces which are subject to the distributionrelations: ARID(Γ) := { M ∈ ARI(Γ) | i N ( M ) = m N ( M ) for all N (cid:62) with N (cid:12)(cid:12) | Γ |} , ARID(Γ) al / al := ARID(Γ) ∩ ARI(Γ) al / al . Proposition 1.31. The Q -linear subspace ARID(Γ) forms a filtered Lie subalgebraof ARI(Γ) under the ari u -bracket.Proof. First we prove that m N is a Lie algebra homomorphism, that is, m N (ari u ( A, B )) = ari u ( m N ( A ) , m N ( B )) for A, B ∈ ARI(Γ) . Let m (cid:62) . We have m N ( A × B ) m ( x m ) = (cid:88) τ Ni = σ i (cid:54) i (cid:54) m ( A × B ) m (cid:0) Nx , ..., Nx m τ , ..., τ m (cid:1) = (cid:88) τ Ni = σ i (cid:54) i (cid:54) m m (cid:88) j =0 A i Ä Nx , ..., Nx j τ , ..., τ j ä B m − i Ä Nx j +1 , ..., Nx m τ j +1 , ..., τ m ä = m (cid:88) j =0 (cid:88) τ Ni = σ i (cid:54) i (cid:54) j A i Ä Nx , ..., Nx j τ , ..., τ j ä (cid:88) τ Ni = σ i j +1 (cid:54) i (cid:54) m B m − i Ä Nx j +1 , ..., Nx m τ j +1 , ..., τ m ä = m (cid:88) j =0 m N ( A ) i Ä x , ..., x j σ , ..., σ j ä m N ( B ) m − i Ä x j +1 , ..., x m σ j +1 , ..., σ m ä = m N ( A × B ) m ( x m ) . Similarly, we have m N (arit u ( B )( A )) m ( x m )= (cid:88) ν Ni = σ i (cid:54) i (cid:54) m arit( B )( A ) m (cid:0) Nx , ..., Nx m ν , ..., ν m (cid:1) = (cid:88) ν Ni = σ i (cid:54) i (cid:54) m (cid:88) (cid:54) k (cid:54) l ARID(Γ) , we have ARID(Γ) = (cid:84) N (cid:12)(cid:12) | Γ | ker( i N − m N ) . Therefore ARID(Γ) forms a Lie subalgebra under the ari u -bracket. (cid:3) Corollary 1.32. The Q -linear subspace ARID(Γ) al / al forms a filtered Lie subalge-bra of ARI(Γ) under the ari u -bracket.Proof. It is a direct consequence of Proposition 1.21 and Proposition 1.31. (cid:3) Kashiwara-Vergne Lie algebra We introduce the Kashiwara-Vergne bigraded Lie algebra lkrv (Γ) •• associatedwith 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 definingconditions of Kashiwara-Vergne graded Lie algebra associated with a finite abeliangroup Γ (cf. Definition 2.1) and explain its mould theoretical interpretation inTheorem 2.15.Let L = ⊕ w (cid:62) 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 generatedby Lie monomials whose total degree is w . Occasionally we regard L as a bigradedLie algebra L •• = ⊕ w,d L w,d , where L w,d is the Q -linear space generated by Liemonomials whose weight (the total degree) is w and depth (the degree with respectto all y σ ) is d . We encode L with a structure of filtered graded Lie algebra by thefiltration Fil d D L w := ⊕ N (cid:62) d L w,N for d > and denote the associated bigraded Liealgebra by gr D L = ⊕ w,d gr d D L w with gr d D L w = Fil d D L w / Fil d +1 D L w .The N + 1 -variable non-commutative polynomial algebra A = Q (cid:104) x, y σ ; σ ∈ Γ (cid:105) is regarded as the universal enveloping algebra of L and is encoded with theinduced degree. Similarly A is encoded with a structure of bigraded algebra; A •• = ⊕ w,d A w,d . By putting Fil d D A w := ⊕ N (cid:62) d A w,N for d > , we also encode A with a structure of filtered graded algebra. We define the action of τ ∈ Γ on A (hence of L ) by τ ( x ) = x and τ ( y σ ) = y τσ . For any h ∈ A , we denote h = h x x + (cid:88) α h y α y α = xh x + (cid:88) α y α h y α . We denote π Y to be the composition of the natural projection and inclusion: π Y : A (cid:16) A / A · x (cid:39) Q ⊕ ( ⊕ σ ∈ Γ A y σ ) (cid:44) → A and the Q -linear isomorphism q on A defined by q ( x e y σ x e y σ · · · x e r − y σ r x e r ) = x e y σ x e y σ σ − · · · x e r − y σ r σ − r − x e r (cf. [R02]). The Q -linear endomorphism anti : A → A is the palindrome (backwards-writing) operator in A w (cf. [Sch12, Definition 1.3]).We put Cyc( A ) to be Q -linear space generated by cyclic words of A and tr : A (cid:16) Cyc( A ) to be the trace map, the natural projection to Cyc( A ) (cf. [AT]). ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 19 Definition 2.1. We define the graded Q -linear space krv (Γ) • = ⊕ w> krv (Γ) w ,where its degree w -part krv (Γ) w is defined to be the set of Lie elements F ∈ L w such that there exists G = G ( F ) in L w with [ x, G ] + (cid:88) τ ∈ Γ [ y τ , τ ( F )] = 0 , (KV1) tr ◦ q ◦ π Y ( F ( z ; ( y σ ))) = 0 (KV2)with z = − x − (cid:80) σ ∈ Γ y σ .We note that such G = G ( F ) uniquely exists when w > . For d (cid:62) , we put Fil d D krv (Γ) w to be the subspace of krv (Γ) w consisting of F ∈ Fil d D L w . By (KV2), krv (Γ) w = Fil D krv (Γ) w . Lemma 2.2. Assume that Γ = { e } . Let F ∈ L w satisfying (KV1) with w (cid:62) .Then (KV2) for F is equivalent to (2.1) tr( G x x + F y y ) = 0 . Proof. By tr ◦ q ◦ π Y ( F ( z, y )) = tr ◦ π Y ( F ( z, y )) = tr( − F x ( z, y ) y + F y ( z, y ) y ) , thecondition (KV2) is equivalent to tr(( F y − F x ) y ) = 0 . By (KV1), we have Gx − xG = yF − F y . So F y = F y and G y = F x . By F ∈ L w , F x = ( − w − anti( F x ) and F y = ( − w − anti( F y ) . By G ∈ L w , we have tr( G x x + G y y ) = 0 . Therefore tr(( F y − F x ) y ) = ( − w − tr(( F y − F x ) y ) = ( − w − tr( F y y − G y y ) = ( − w − tr( F y y + G x x ) , whence we get the claim (cid:3) Remark 2.3. Since (2.1) agrees with original defining condition (KV2) in [AT],we see that our krv (Γ) • with Γ = { e } recovers the original Kashiwara-Vergne Liealgebra denoted by krv in [AT] and the depth>1-part of krv in [RS, Definition 3].We do not know how their Lie algebras krv n +1 ( n (cid:62) ) in [AT] and also krv n +1 in[AKKN] are related to our krv (Γ) • . It is also not clear if our krv (Γ) • forms a Liealgebra or not.Let h ∈ A w be a degree w homogeneous polynomial with h = (cid:80) wr =0 h r and(2.2) h r = (cid:88) ( σ ,...,σ r ) ∈ Γ ⊕ r (cid:88) ( e ,...,e r ) ∈E rw a ( h : e ,...,e r σ ,...,σ r ) x e y σ · · · y σ r x e r ∈ A w,r where E rw = { ( e , . . . , e r ) ∈ N r +10 | (cid:80) ri =0 e i = w − r } . Definition 2.4. By following [Sch12, Appendix A], we associate a mould ma h = (ma h , ma h , ma h , . . . , ma wh , , , . . . ) ∈ M ( F ; Γ) which is defined by ma rh = { ma rh ( u ,...,u r σ ,...,σ r ) } ( σ ,...,σ r ) ∈ Γ ⊕ r with ma rh ( u ,...,u r σ ,...,σ r ) = vimo rh ( ,u ,u + u ,...,u + ··· + u r σ ,...,σ r ) , vimo rh ( z ,...,z r σ ,...,σ r ) = (cid:88) ( e ,...,e r ) ∈E rw a ( h : e ,...,e r σ − ,...,σ − r ) z e z e z e · · · z e r r . We start with the following technical lemma which is required to our later argu-ments. Lemma 2.5. When h ∈ L w,r , we have (2.3) vimo rh ( z ,...,z r σ ,...,σ r ) = vimo rh ( ,z − z ,...,z r − z σ ,...,σ r ) , (2.4) mantar ◦ ma rh = ma rh . Proof. The proof of (2.3) can be done by induction on degree in the same way tothe arguments in [Sch12] p.71. Assume h ∈ L w,r is given by [ f, g ] for some f and g ∈ L with depth s and t respectively. Then by definition, we have vimo rh ( z ,...,z r σ ,...,σ r ) = vimo sf ( z ,...,z s σ ,...,σ s )vimo tg ( z s ,...,z s + t σ s +1 ,...,σ s + t ) − vimo tg ( z ,...,z t σ ,...,σ t )vimo sf ( z t ,...,z s + t σ t +1 ,...,σ s + t ) . While by our induction assumption we have vimo rh ( ,z − z ,...,z r − z σ ,...,σ r )= vimo sf ( ,z − z ,...,z s − z σ ,...,σ s )vimo tg ( z s − z ,...,z s + t − z σ s +1 ,...,σ s + t ) − vimo tg ( ,z − z ,...,z t − z σ ,...,σ t )vimo sf ( z s − z ,...,z s + t − z σ t +1 ,...,σ s + t )= vimo sf ( z ,...,z s σ ,...,σ s )vimo tg ( z s ,...,z s + t σ s +1 ,...,σ s + t ) − vimo tg ( z ,...,z t σ ,...,σ t )vimo sf ( z t ,...,z s + t σ t +1 ,...,σ s + t )= vimo rh ( z ,...,z r σ ,...,σ r ) . The equation (2.4) is a formal generalization of [Sch12, Lemma A.2]. We give ashort proof below. Since h ∈ L w , we have h = ( − w − anti( h ) . Thus by definition,we have vimo rh ( z ,...,z r σ ,...,σ r ) = ( − w − vimo rh ( z r ,...,z σ r ,...,σ ) . Therefore ma rh ( z ,...,z r σ ,...,σ r ) = vimo rh ( ,z ,z + z ,...,z + ··· + z r σ ,...,σ r ) = ( − w − vimo rh ( z + ··· + z r ,...,z + z ,z , σ r ,...,σ )= ( − r − vimo rh ( − z −···− z r ,..., − z − z , − z , σ r ,...,σ ) = ( − r − vimo rh ( ,z r ,z r + z r − ,...,z r + ··· + z σ r ,...,σ )= ( − r − ma rh ( z r ,...,z σ r ,...,σ ) = mantar ◦ ma rh ( z ,...,z r σ ,...,σ r ) . Here in the third equality we use that vimo rh is homogeneous with degree w − r andthe fourth equality is by (2.3). (cid:3) Definition 2.6. Let tder be the set of tangential derivation of L , the derivation D { F σ } σ ,G of L defined by x (cid:55)→ [ x, G ] and y σ (cid:55)→ [ y σ , F σ ] for some F σ , G ∈ L . Itforms a Lie algebra by the bracket(2.5) [ D { F (1) σ } ,G (1) , D { F (2) σ } ,G (2) ] = D { F (1) σ } ,G (1) ◦ D { F (2) σ } ,G (2) − D { F (2) σ } ,G (2) ◦ D { F (1) σ } ,G (1) . The action of Γ on L induces the Γ -action on tder . We denote its invariant partby tder Γ . We mean sder to be set of special derivations, tangential derivations suchthat D { F σ } ,G ( x + (cid:80) σ y σ ) = 0 and sder Γ to be its intersection with tder Γ , both ofwhich forms a Lie subalgebra of tder . We put mt := ⊕ ( w,d ) (cid:54) =(1 , L w,d . It forms aLie algebra by the bracket(2.6) { f , f } = D { σ ( f ) } , ( f ) − D { σ ( f ) } , ( f ) + [ f , f ] , in other words, D { σ ( { f ,f } ) } , := [ D { σ ( f ) } , , D { σ ( f ) } , ] .We occasionally regard mt as a Lie subalgebra of tder by f (cid:55)→ D { σ ( f ) } , . We notethat 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 Fil d D sder Γ w = { D { σ ( F ) } ,G ∈ sder Γ (cid:12)(cid:12) F ∈ Fil d D L w } and Fil d D mt = { D { σ ( F ) } , ∈ mt (cid:12)(cid:12) F ∈ Fil d D L w } . The following is required in the next section. ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 21 Lemma 2.7. We have a natural graded Lie algebra homomorphism (2.7) res : gr D ( sder Γ ) → mt sending D { τ ( F ) } ,G (cid:55)→ D { τ ( ¯ F ) } , .Proof. Let D i = D { τ ( F i ) } ,G i ∈ Fil d i D sder Γ w i for i = 1 , . Put D = [ D , D ] . Since itbelongs to sder Γ , D is expressed as D { τ ( F ) } ,G for some F ∈ L and G ∈ L .By (KV1), we have τ ( F i ) ∈ Fil d i D L w i , G i ∈ Fil d i +1 D L w i ( i = 1 , ). Since F iscalculated to be F = D { τ ( F ) } ,G ( F ) − D { τ ( F ) } ,G ( F ) + [ F , F ] by (2.5), we see that F ∈ Fil d + d D L w + w . The residue class ¯ F ∈ gr d + d D L w + w is calculated as(2.8) ¯ F = D { τ ( ¯ F ) } , ( ¯ F ) − D { τ ( ¯ F ) } , ( ¯ F ) + [ ¯ F , ¯ F ] , where ¯ F ∈ gr d D L w and ¯ F ∈ gr d D L w are the residue classes of F and F . There-fore our map is a Lie algebra homomorphism. (cid:3) Proposition 2.8. The map ma sending h → ma h induces a filtered graded Liealgebra isomorphism (2.9) ma : mt (cid:39) ARI(Γ) fin , polal where ARI(Γ) fin , polal is the Lie algebra (cf. Lemma 1.13) equipped with the ari u -bracket.Proof. It can be proved completely in a same way to that of [Sch15, Theorem3.4.2]. (cid:3) We prepare the following technical lemma which is required to the proof of areformulation of (KV1) in Lemma 2.10. Lemma 2.9. Let H ∈ L w with w (cid:62) . Assume that H has no words starting withany y σ and ending in any y τ . Then there exists G ∈ L w − 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 (cid:55)→ and y σ (cid:55)→ and the Q -linear en-domorphism sec of A by sec( h ) := (cid:80) i (cid:62) − i i ! ∂ ix ( h ) x i for h ∈ A . Let us write H = H x x + (cid:80) σ H y σ y σ . Then by our assumption, we have P ∈ A such that xP = (cid:80) σ H y σ y σ . Then by [R02] Proposition 4.2.2 and ∂ ix ( xP ) = i∂ i − x ( P ) + x∂ ix ( P ) , wehave H = sec( (cid:88) σ H y σ y σ ) = sec( xP ) = (cid:88) i (cid:62) ( − i i ! ∂ ix ( xP ) x i = (cid:88) i (cid:62) ( − i ( i − ∂ i − x ( P ) x i + x (cid:88) i (cid:62) ( − i i ! ∂ ix ( P ) x i = − sec( P ) x + x sec( P ) = [ x, sec( P )] . It remains to show that G = sec( P ) is in L , which follows from exactly the samearguments to the last half of the proof of [Sch12, Proposition 2.2]. (cid:3) Let F = F ( x ; ( y σ )) ∈ L . As in [Sch12], we put(2.10) f ( x ; ( y σ )) = F ( z ; ( y σ )) and ˜ f ( x ; ( y σ )) = f ( x ; ( − y σ )) with z = − x − (cid:80) σ y σ .The following generalizes the equivalence between (i) and (v) in [Sch12, Theorem2.1]. Lemma 2.10. Let F ∈ L w with w (cid:62) . Then saying (KV1) for F is equivalent tosaying that (2.11) { ˜ f y γ + ˜ f x } = ( − w − γ · anti { ˜ f y γ − + ˜ f x } for all γ ∈ Γ .Proof. Firstly, we show that (KV1) for F is equivalent to α ( F ) y β = β ( F ) y α for any α, β ∈ Γ . Set H = (cid:80) σ [ y σ , σ ( F )] . Then we have(2.12) H = (cid:88) α,β y α (cid:8) α ( F ) y β − β ( F ) y α (cid:9) y β + (cid:88) α y α α ( F ) x x − (cid:88) β xβ ( F ) x y β . Assume (KV1) for F . Then(2.13) H = Gx − xG. So H has no words starting with any y σ and ending in any y τ . By (2.12), we have α ( F ) y β = β ( F ) y α . Conversely assume α ( F ) y β = β ( F ) y α . Then H has no wordsstarting with any y σ and ending in any y τ . By Lemma 2.9, there is a G ∈ L w suchthat H = [ G, x ] . Whence we get (KV1).Secondly, by α ( F ) ∈ L w , we have α ( F ) y β = ( − w − anti( α ( F ) y β ) . Therefore α ( F ) y β = β ( F ) y α is equivalent to α ( F ) y β = ( − w − anti( β ( F ) y α ) .Lastly, by (2.10) we have f y α ( z ; ( y σ )) − f x ( z ; ( y σ )) = F y α . Hence α ( F ) y β = α ( F y α − β ) = α ¶ f y α − β ( z ; ( y σ )) − f x ( z ; ( y σ )) © Whence α ( F ) y β = ( − w − anti( β ( F ) y α ) is equivalent to α ¶ f y α − β ( z ; ( y σ )) − f x ( z ; ( y σ )) © = ( − w − anti Ä β ¶ f y αβ − ( z ; ( y σ )) − f x ( z ; ( y σ )) ©ä from which our claim follows. (cid:3) The following reformulation is suggested by the arguments in [Sch12, AppendixA]. Lemma 2.11. Let ˜ f ∈ L w with w > . Then (2.11) for all γ ∈ Γ is equivalent tothe following senary relation (cf. [Ec11, (3.64)] ) (2.14) teru( M ) r = push ◦ mantar ◦ teru ◦ mantar( M ) r for (cid:54) r (cid:54) w with M = ma ˜ f .Proof. Because M = ma ˜ f is mantar invariant for ˜ f ∈ L w by (2.4), the senaryrelation (2.14) is equivalent to(2.15) swap ◦ teru( M ) r = swap ◦ push ◦ mantar ◦ teru( M ) r . It is because it consists of 6 terms, 3 terms on each hand sides. ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 23 For simplicity, we denote σ i · · · σ j by σ i,j for (cid:54) i (cid:54) j (cid:54) r . By definition, its lefthand side is calculated to be vimo r ˜ f Ä , v r , . . . , v , v σ ,r , . . . , σ , , σ ä (2.16) + 1 v − v (cid:110) vimo r − f Ä , v r , . . . , v , v σ ,r , . . . , σ , , σ , ä − vimo r − f Ä , v r , . . . , v , v σ ,r , . . . , σ , , σ , ä (cid:111) . By vimo r ˜ f r (cid:0) z ,...,z r σ ,...,σ r (cid:1) = ( − w − r vimo r ˜ f r (cid:0) − z ,..., − z r σ ,...,σ r (cid:1) and definition, its right handside is calculated to be ( − w − (cid:104) vimo r ˜ f (cid:16) , v − v , . . . , v r − v , − v , v − v σ , , . . . , σ ,r − , σ ,r , σ − (cid:17) (2.17) + 1 v (cid:110) vimo r − f Ä , v − v , . . . , v r − v , v − v σ , , . . . , σ ,r − , σ ,r ä − vimo r − f Ä , v − v , . . . , v r − v , − v σ , , . . . , σ ,r − , σ ,r ä (cid:111) ò . By vimo r ˜ f r (cid:0) z ,...,z r σ ,...,σ r (cid:1) = vimo r ˜ f r (cid:0) ,z − z ,...,z r − z σ ,...,σ r (cid:1) in (2.3) and the change of variables z = − v , z = v r − v , . . . , z r − = v − v and γ = σ , τ = σ ,r , . . . , τ r − = σ , ,the formula (2.16) is equal to vimo r ˜ f r Ä z ,...,z r − , τ ,...,τ r − ,γ ä + 1 z r − (cid:110) vimo r − f r − Ä z ,...,z r − τ ,...,τ r − ä − vimo r − f r − Ä z ,...,z r − , τ ,...,τ r − ,τ r − ä (cid:111) , (2.18)and (2.17) is equal to ( − w − î vimo r ˜ f r Ä z r − ,...,z , γ − τ r − ,...,γ − τ ,γ − ä (2.19) + 1 z (cid:110) vimo r − f r − Ä z r − ,...,z ,z γ − τ r − ,...,γ − τ ä − vimo r − f r − Ä z r − ,...,z , γ − τ r − ,...,γ − τ ä (cid:111) ò . Whence (2.14) is equivalent to (2.18)=(2.19) for (cid:54) r (cid:54) w and τ , . . . , τ r − , γ ∈ Γ .On the other hand, (2.11) for all γ ∈ Γ is equivalent to vimo r ˜ f ryγ (cid:0) z ,...,z r τ ,...,τ r (cid:1) + vimo r ˜ f rx (cid:0) z ,...,z r τ ,...,τ r (cid:1) (2.20) = ( − w − ï vimo rγ · anti( ˜ f ryγ − ) (cid:0) z ,...,z r τ ,...,τ r (cid:1) + vimo rγ · anti( ˜ f rx ) (cid:0) z ,...,z r τ ,...,τ r (cid:1) ò for γ ∈ Γ and (cid:54) r (cid:54) w . By definition, we have vimo r +1( ˜ f yγ ) r y γ Ä z ,...,z r +1 τ ,...,τ r +1 ä = δ τ r +1 ,γ vimo r +1˜ f r +1 (cid:0) z ,...,z r , τ ,...,τ r ,γ (cid:1) where δ τ r +1 ,γ := ß τ r +1 = γ ) , otherwise ) . So by using this, we get vimo r ˜ f ryγ (cid:0) z ,...,z r τ ,...,τ r (cid:1) = vimo r +1( ˜ f yγ ) r y γ (cid:0) z ,...,z r , τ ,...,τ r ,γ (cid:1) = vimo r +1˜ f r +1 (cid:0) z ,...,z r , τ ,...,τ r ,γ (cid:1) . (2.21)By ˜ f r = ( ˜ f x ) r x + (cid:80) γ ∈ Γ ( ˜ f y γ ) r − y γ and vimo r ( ˜ f x ) r x (cid:0) z ,...,z r τ ,...,τ r (cid:1) = z r · vimo r ˜ f rx (cid:0) z ,...,z r τ ,...,τ r (cid:1) ,we have vimo r ˜ f rx (cid:0) z ,...,z r τ ,...,τ r (cid:1) = 1 z r ¶ vimo r ˜ f r (cid:0) z ,...,z r τ ,...,τ r (cid:1) − vimo r ˜ f r Ä z ,...,z r − , τ ,...,τ r − ,τ r ä© . (2.22) Here ˜ f rx and ˜ f ry mean ( ˜ f x ) r and ( ˜ f y ) r . Hence, by (2.21) and (2.22), the left hand side of (2.20) is calculated to be vimo r +1˜ f r +1 (cid:0) z ,...,z r , τ ,...,τ r ,γ (cid:1) + 1 z r ¶ vimo r ˜ f r (cid:0) z ,...,z r τ ,...,τ r (cid:1) − vimo r ˜ f r Ä z ,...,z r − , τ ,...,τ r − ,τ r ä© , (2.23)and the left hand side of (2.20) is calculated to be ( − w − (cid:104) vimo r +1˜ f r +1 Ä z r ,...,z , γ − τ r ,...,γ − τ ,γ − ä (2.24) + 1 z ¶ vimo r ˜ f r Ä z r ,...,z ,z γ − τ r ,...,γ − τ ä − vimo r ˜ f r Ä z r ,...,z , γ − τ r ,...,γ − τ ä©ò . So (2.11) for all γ ∈ Γ is equivalent to (2.23)=(2.24) for (cid:54) r + 1 (cid:54) w and τ , . . . , τ r , γ ∈ Γ . This is nothing but (2.18)=(2.19) for r − instead of r . Thereforewe get the equivalence between (2.11) and (2.14). (cid:3) Next we consider the condition (KV2). Lemma 2.12. Let F ∈ L w . Then (KV2) for F is equivalent to (2.25) (cid:88) i ∈ Z /r Z a ( q ◦ π Y ( f ) : e i ,e i +1 ,...,e i + r − , σ i +1 ,σ i +2 ,...,σ i + r ) = 0 for each (cid:54) r (cid:54) w , ( σ , . . . , σ r ) ∈ Γ r and ( e , . . . , e r ) ∈ E rw with f as in (2.10) .Proof. It is immediate that (KV2) for F is equivalent to(2.26) tr( q ◦ π Y ( f )) = 0 . For W = u · · · u w ∈ A w with u i = x or y σ ( σ ∈ Γ ), we define its cyclicpermutation by cy ( W ) := u · · · u w u ∈ A w . We put c ( W ) to be the number of itscycles. It divides w . Then the natural pairing in Cyc( A ) is calculated by the onein A as follows: When W = x e y σ · · · x e r − y σ r , we have (cid:104) tr { q ◦ π Y ( f ) (cid:12)(cid:12)(cid:12) tr( W ) (cid:105) Cyc( A ) = (cid:10) q ◦ π Y ( f ) (cid:12)(cid:12)(cid:12) c ( W ) − (cid:88) i =0 cy i ( W ) (cid:11) A = c ( W ) w (cid:10) q ◦ π Y ( f ) (cid:12)(cid:12) w − (cid:88) i =0 cy i ( W ) (cid:11) A = c ( W ) w (cid:88) i ∈ Z /r Z a ( q ◦ π Y ( f ) : e i ,...,e i + r − , σ i +1 ,...,σ i + r ) . So we get the claim. (cid:3) Lemma 2.13. Let f ∈ L w . Then (2.25) for f is equivalent to the pus-neutrality (1.8) for swap(ma ˜ f ) , i.e. (2.27) (cid:88) i ∈ Z /r Z pus i ◦ swap( M ) r (cid:0) σ ,...,σ r v ,...,v r (cid:1) = 0 with M = ma ˜ f for all (cid:54) r (cid:54) w and σ i ∈ Γ ( (cid:54) i (cid:54) r ).Proof. We decompose ˜ f = (cid:80) r ˜ f r and describe ˜ f r ∈ L w,r as in (2.2). Then ma r ˜ f ( u ,...,u r σ ,...,σ r ) = (cid:88) ( e ,...,e r ) ∈E rw e =0 a ( ˜ f : e ,...,e r σ − ,...,σ − r ) u e ( u + u ) e · · · ( u + · · · + u r ) e r . By ˜ f r ∈ L w,r , we have a ( ˜ f : e ,...,e r σ ,...,σ r ) = ( − w +1 a ( ˜ f : e r ,...,e σ r ,...,σ ) ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 25 because anti( ˜ f r ) = ( − w +1 ˜ f r . So swap ◦ ma r ˜ f ( σ ,...,σ r v ,...,v r ) = (cid:88) ( e ,...,e r ) ∈E rw e =0 a ( ˜ f : e ,e ,...,e r ( σ ··· σ r ) − ,..., ( σ σ ) − ,σ − ) v e r v e r − · · · v e r = ( − w +1 (cid:88) ( e ,...,e r ) ∈E rw e =0 a ( ˜ f : e r ,...,e ,e σ − , ( σ σ ) − ,..., ( σ ··· σ r ) − ) v e r v e r − · · · v e r = ( − w +1 (cid:88) ( e ,...,e r ) ∈E rw e r =0 a ( ˜ f : e ,...,e r − ,e r σ − , ( σ σ ) − ,..., ( σ ··· σ r ) − ) v e v e · · · v e r − r = ( − w + r +1 (cid:88) ( e ,...,e r ) ∈E rw e r =0 a ( f : e ,...,e r − ,e r σ − , ( σ σ ) − ,..., ( σ ··· σ r ) − ) v e v e · · · v e r − r = ( − w + r +1 (cid:88) ( e ,...,e r ) ∈E rw e r =0 a ( q ◦ π Y ( f ) : e ,...,e r − , σ − ,σ − ,...,σ − r ) v e v e · · · v e r − r . Here in the last equality we use a ( q ◦ π Y ( h ) : e ,...,e r − , τ ,...,τ r ) = a ( h : e ,...,e r − , τ ,τ τ ,...,τ ··· τ r ) for any h ∈ π ( A ) . Therefore we obtain (cid:88) i ∈ Z /r Z pus i ◦ swap ◦ ma r ˜ f ( σ ,...,σ r v ,...,v r )= ( − w + r +1 (cid:88) i ∈ Z /r Z (cid:88) ( e ,...,e r ) ∈E rw e r =0 a ( q ◦ π Y ( f ) : e i ,...,e i + r − , σ − i +1 ,...,σ − i + r ) v e v e · · · v e r − r . It is immediate to see that it is equivalent to (2.25). (cid:3) The following definition of the mould version of krv • is suggested by our previouslemmas. Definition 2.14. We define the Q -linear space ARI(Γ) sena / pusnu to be the subsetof moulds M in ARI which satisfy the senary relation (2.14) and whose swap satisfythe pus-neutrality (2.27).The Q -linear space ARI(Γ) sena / pusnu is filtered by the depth filtration { Fil m D } m .We note that Fil D ARI(Γ) sena / pusnu ∩ ARI(Γ) fin , polal is identified with the depth >1-part of ARI polal+sen ∗ circconst in [RS, Proposition 28] when Γ = { e } . It is becausetheir formula (79), actually (81), agrees with (2.14) for a mould in ARI(Γ) fin , polal byLemma 3.7.There is an embedding of Q -linear space(2.28) gr D ARI(Γ) sena / pusnu (cid:44) → ARI(Γ) push / pusnu that is, the associated graded quotient gr D ARI(Γ) sena / pusnu of the filtered Q -linearspace ARI(Γ) sena / pusnu is embedded to ARI(Γ) push / pusnu introduced in Definition1.23 because the push-invariance (1.4) is associated graded with (2.14) by gr D (teru) =id and mantar ◦ mantar = id . Theorem 2.15. The map sending F ∈ A (cid:55)→ ma ˜ f ∈ M ( F ; Γ) induces an isomor-phism of filtered Q -linear spaces (2.29) krv (Γ) • (cid:39) ARI(Γ) sena / pusnu ∩ ARI(Γ) fin , polal . Proof. The restriction of our map decomposes as(2.30) F ∈ krv (Γ) • (cid:55)→ ˜ f ∈ mt (cid:55)→ ma ˜ f ∈ ARI(Γ) fin , polal . By Proposition 2.8, we see that it gives an isomorphism mt (cid:39) ARI(Γ) fin , polal as(actually filtered) Q -linear spaces. Thus we get the claim by our previous lemmas. (cid:3) Remark 2.16. When Γ = { e } , the above isomorphism (2.29) can be recovered bythe composition of the isomorphisms (29), (71) and Proposition 28 in [RS].The authors are not sure if the bigger space ARI(Γ) sena / pusnu is equipped witha structure of Lie algebra under the ari u -bracket or not although we show that arelated space ARI(Γ) push / pusnu forms a Lie algebra in Theorem 1.28.2.2. Kashiwara-Vergne bigraded Lie algebra. Based on our arguments in theprevious subsection, we introduce a Γ -variant lkrv (Γ) •• of the Kashiwara-Vergnebigraded Lie algebra lkrv ([RS]) in Definition 2.17 and give a mould theoreticalinterpretation in Theorem 2.22. As a corollary we show that it forms a Lie algebrain Theorem 2.23. Definition 2.17. Kashiwara-Vergne bigraded Lie algebra is defined to be the bi-graded Q -linear space lkrv (Γ) •• = ⊕ w> ,d> lkrv (Γ) w,d , where lkrv (Γ) w,d is the Q -linear space consisting of ¯ F ∈ gr d D L w whose lift F ∈ Fil d D L w satisfies the followingrelations, ‘the leading-terms’ of (KV1) and (KV2), [ x, G ] + (cid:88) τ ∈ Γ [ y τ , τ ( F )] ≡ d +2 D L w +1 , (LKV1) tr ◦ q ◦ π Y ( F ) ≡ d +1 D A w ) (LKV2)for a certain G = G ( F ) ∈ L w .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 , we have lkrv w,d = { } for d = 1 . Remark 2.18. When Γ = { e } , our definition of lkrv (Γ) •• does not agree with thatof lkv in [RS, Definition 5 and 10], which is constructed as if a Lie polynomial versionof ARI(Γ) push / pusnu . However their depth>1-parts are eventually isomorphic viaTheorem 2.22.By definition, there is an inclusion of bigraded Q -linear spaces(2.31) gr D krv (Γ) • (cid:44) → lkrv (Γ) •• , 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 notknow if it is an isomorphism. It looks that there is a slip on the isomorphism on [RS, Proposition 28] because its righthand side is completed by depth while its left hand side is not. ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 27 For F ∈ Fil d D L w , we put f to be the element in L w,d corresponding to ( − w − d ¯ F ∈ gr m D L w under the natural identification gr d D L w (cid:39) L w,d . By abuse of notation,(2.32) f = ( − w − d ¯ F . We write f = f x x + (cid:80) σ f y σ y σ . We also put ˜ f = f ( x, ( − y σ ) σ ) and write ˜ f = ˜ f x x + (cid:80) σ ˜ f y σ y σ .The following is a bigraded variant of Lemma 2.10. Lemma 2.19. Let F ∈ Fil d D L w for w (cid:62) . Then (LKV1) for F is equivalent to (2.33) ˜ f y γ = ( − w − γ (anti(˜ f y γ − )) for any γ ∈ Γ .Proof. The proof goes similarly to Lemma 2.10.Firstly, we show that (LKV1) for F is equivalent to α ( F ) y β ≡ β ( F ) y α mod Fil d D L w − .Set H = (cid:80) σ [ y σ , σ ( F )] ∈ Fil d +1 D L w +1 .Assume (LKV1) for F . Then H ≡ Gx − xG mod Fil d +2 D L w +1 . So the depth ( d + 1) -part of H has no words starting and ending in any y σ . By (2.12), we have α ( F ) y β ≡ β ( F ) y α mod Fil d D L w − .Conversely assume α ( F ) y β ≡ β ( F ) y α mod Fil d D L w − . Then the depth ( d + 1) -part of H has no words starting and ending in any y σ . By Lemma 2.9, there isa G ∈ L w which express this part as [ G, x ] , i.e. H ≡ [ G, x ] mod Fil d +2 D L w +1 .Whence we get (LKV1).Secondly, by α ( F ) ∈ L , we have α ( F ) y β = ( − w − anti( α ( F ) y β ) . Therefore α ( F ) y β ≡ β ( F ) y α mod Fil d D L w − is equivalent to α ( F ) y β ≡ ( − w − anti( β ( F ) y α ) mod Fil d D L w − . Lastly, by (2.32) and α ( F ) y β = α ( F y α − β ) , it is equivalent to α ( f y α − β ) =( − w − anti( β ( f y αβ − )) , so for ˜ f y . (cid:3) The following might be regarded as a bigraded variant of Lemma 2.11. Lemma 2.20. Let ˜ f ∈ L w,d with w > . Then (2.33) for any γ ∈ Γ is equivalentto push -invariance (1.4) for M = ma ˜ f .Proof. The proof goes similarly to Lemma 2.11. The condition (2.33) for γ ∈ Γ isreformulated to vimo r ˜ f ryγ (cid:0) z ,...,z r τ ,...,τ r (cid:1) = ( − w − vimo rγ · anti(˜ f ryγ − ) (cid:0) z ,...,z r τ ,...,τ r (cid:1) for (cid:54) r (cid:54) w , which is equivalent to(2.34) vimo r +1˜ f r +1 (cid:0) z ,...,z r , τ ,...,τ r ,γ (cid:1) = ( − w − vimo r +1˜ f r +1 Ä z r ,...,z , γ − τ r ,...,γ − τ ,γ − ä for (cid:54) r + 1 (cid:54) w .On the other hand, (1.4) can be shown to be equivalent to vimo r ˜ f r Ä z ,...,z r − , τ ,...,τ r − ,γ ä = ( − w − vimo r ˜ f r Ä z r − ,...,z , γ − τ r − ,...,γ − τ ,γ − ä for (cid:54) r (cid:54) w , which turns to be equivalent to (2.34). (cid:3) The following is a bigraded variant of Lemma 2.13. Lemma 2.21. Then (LKV2) for F ∈ Fil d D L w is equivalent to the pus -neutrality (1.8) for M = swap(ma ˜ f ) , i.e. (2.27) for all r (cid:62) ,Proof. We note that actually only the terms for r = d contributes in the aboveequation. Decompose ˜ f as in (2.2). Then by the arguments in the proof of Lemma2.13, (2.27) is equivalent to (cid:88) i ∈ Z /r Z a ( q ◦ π Y ( f ) : e i ,...,e i + r − , σ − i +1 ,...,σ − i + r ) = 0 for all ( σ , . . . , σ r ) ∈ Γ r and ( e , . . . , e r − , ∈ E rw . It is nothing but tr ◦ q ◦ π Y ( f ) = 0 , which is equivalent to (LKV2). (cid:3) Theorem 2.22. The map sending ¯ F ∈ gr D L (cid:55)→ ma ˜ f ∈ M ( F ; Γ) induces an iso-morphism of bigraded Q -linear spaces lkrv (Γ) •• (cid:39) ARI(Γ) push / pusnu ∩ ARI(Γ) fin , polal . Proof. Our map decomposes as(2.35) ¯ F ∈ lkrv (Γ) •• (cid:55)→ ˜ f ∈ mt (cid:55)→ ma ˜ f ∈ ARI(Γ) fin , polal . The first map is injective and the second one is isomorphic by Proposition 2.8. Ourclaim follows by our previous lemmas. (cid:3) As a generalization of [RS, Proposition 1], we obtain the following. Theorem 2.23. The space lkrv (Γ) •• forms a bigraded Lie algebra under the bracket (2.6) .Proof. As for the morphism (2.35), the second map is Lie algebra homomorphismby Proposition 2.8. It is easy to see that the first map forms a Lie algebra homo-morphism when we encode lkrv (Γ) •• with the bracket (2.6). Thus our claim followsbecause it is shown that ARI(Γ) push / pusnu forms a Lie algebra by Theorem 1.28 andso ARI(Γ) al does by Lemma 1.13. (cid:3) Remark 2.24. By (2.28), (2.31), Theorem 2.15 and Theorem 2.22, we obtain thefollowing commutative diagram of Lie algebras.(2.36) gr D krv (Γ) • (cid:39) (cid:47) (cid:47) (cid:127) (cid:95) (cid:15) (cid:15) gr D (ARI(Γ) sena / pusnu ∩ ARI(Γ) fin , polal ) (cid:127) (cid:95) (cid:15) (cid:15) lkrv (Γ) •• (cid:39) (cid:47) (cid:47) ARI(Γ) push / pusnu ∩ ARI(Γ) fin , polal The diagram is commutative because (2.35) is associated graded with (2.30).Similarly to Definition 1.30, we impose a distribution relation on lkrv (Γ) . Definition 2.25. For N (cid:62) and Γ N = { g N ∈ Γ | g ∈ Γ } , we consider the map i N : lkrv (Γ) •• → lkrv (Γ N ) •• which is induced by i N ( x ) = x, i N ( y τ ) = ® y τ when τ ∈ Γ N , when τ (cid:54)∈ Γ N ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 29 and also the map m N : lkrv (Γ) •• → lkrv (Γ N ) •• which is induced by m N ( x ) = N x, m N ( y σ ) = y σ N for σ ∈ Γ . We define the following Q -linear subspace lkrvd (Γ) •• := { ¯ F ∈ lkrv (Γ) •• (cid:12)(cid:12) i N ( ¯ F ) = m N ( ¯ F ) for all N (cid:62) } . 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 thefollowing isomorphism of bigraded Lie algebras lkrvd (Γ) •• (cid:39) ARI(Γ) push / pusnu ∩ ARID(Γ) fin , polal . Proof. By definition we see that both i N and m N form Lie algebra homomorphismsand by Proposition 1.12 ARI(Γ) forms a Lie algebra. Therefore our claim followsfrom the commutativity of the following diagrams and Proposition 1.31: lkrv (Γ) •• (cid:47) (cid:47) i N (cid:15) (cid:15) ARI(Γ) i N (cid:15) (cid:15) lkrv (Γ N ) •• (cid:47) (cid:47) ARI(Γ N ) , lkrv (Γ) •• (cid:47) (cid:47) m N (cid:15) (cid:15) ARI(Γ) m N (cid:15) (cid:15) lkrv (Γ N ) •• (cid:47) (cid:47) ARI(Γ N ) . (cid:3) Dihedral Lie algebra By using mould theoretic interpretations of the bigraded Lie algebra D (Γ) •• ( Γ :a finite abelian group) with a dihedral symmetry and of the Kashiwara-Vergnebigraded Lie algebra lkrv (Γ) •• , we realize an embedding Fil D D (Γ) •• (cid:44) → lkrv (Γ) •• which extends the result of [RS].3.1. Dihedral bigraded Lie algebra. We recall the definition of the dihedral bi-graded Lie algebra D (Γ) •• introduced by Goncharov [G01a] and introduce a relatedLie algebra D (Γ) •• which contains D (Γ) •• in Definitions 3.1 and 3.4.We call an element f = f ( t , . . . , t m +1 ) in Q [ t , . . . , t m +1 ] translation invariant when f ( t , . . . , t m +1 ) = f ( t + c, . . . , t m +1 + c ) for any c ∈ Q . We often denote thisby f ( t : · · · : t m +1 ) . We consider a set of collections(3.1) Z ∼ = { Z ∼ ( g , . . . , g m , g m +1 | t : · · · : t m +1 ) } g ,...,g m ∈ Γ with g m +1 = ( g · · · g m ) − of translation invariant element in Q [ t , . . . , t m +1 ] . For such Z ∼ , we associate ˜ Z ( g : · · · : g m +1 | t , . . . , t m +1 ) := (3.2) Z ∼ ( g − g , g − g , . . . , g − m g m +1 , g − m +1 g | t : t + t : · · · : t + · · · + t m : 0) with any g , . . . , g m +1 ∈ Γ and t + · · · + t m +1 = 0 and also Z ( g : · · · : g m +1 | t : · · · : t m +1 ):= ˜ Z ( g : · · · : g m +1 | t − t m +1 , t − t , . . . , t m − t m − , t m +1 − t m ) (3.3) = Z ∼ ( g − g , g − g , . . . , g − m g m +1 , g − m +1 g | t : t : · · · : t m : t m +1 ) (3.4)with any g , . . . , g m +1 ∈ Γ and any t , . . . , t m +1 . Definition 3.1 ([G01a]) . Set-theoretically the dihedral bigraded Lie algebra meansthe Q -linear bigraded space D (Γ) •• = (cid:77) w,m D (Γ) w,m , where the bidegree ( w, m ) -part D (Γ) w,m is defined to be the set Z ∼ in (3.1) oftranslation invariant elements in Q [ t , . . . , t m +1 ] with total degree w − m satisfying(a). the double shuffle relation , that is,(a-i). the harmonic product (3.5) (cid:88) σ ∈ X p,q Z ∼ ( g σ (1) , . . . , g σ ( m ) , g m +1 | t σ (1) : · · · : t σ ( m ) : t m +1 ) = 0 (a-ii). the shuffle product (3.6) (cid:88) σ ∈ X p,q ˜ Z ( g σ (1) : · · · : g σ ( m ) : g m +1 | t σ (1) , . . . , t σ ( m ) , t m +1 ) = 0 for any p, q (cid:62) with p + q = m .(b). the distribution relation for N ∈ Z such that | N | divides | Γ | ,(3.7) Z ( g : · · · : g m +1 | t : · · · : t m +1 ) = 1 | Γ N | (cid:88) h Ni = g i Z ( h : · · · : h m +1 | N t : · · · : N t m +1 ) except that a constant in t is allowed when m = 1 and g = g . Here | Γ N | is theorder of the N -torsion subgroup of Γ and X p,q is defined by (1.2).(c). Additionally we put(3.8) Z ( e : e | for a technical reason.Its Lie algebra structure is explained in [G01a] §§4-5. In [G01a, Theorem 4.1], itis shown that the double shuffle relation implies the dihedral symmetry relations: Theorem 3.2 ([G01a] Theorem 4.1) . The double shuffle relation implies the dihe-dral symmetry relations, which consist of the cyclic symmetry relation Z ( g : g : · · · : g m +1 | t : t : · · · : t m +1 ) = Z ( g : · · · : g m +1 : g | t : · · · : t m +1 : t ) , the inversion relation (‘the distribution relation for N = − ’) Z ( g : · · · : g m +1 | t : · · · : t m +1 ) = Z ( g − : · · · : g − m +1 | − t : · · · : − t m +1 ) . the reflection relation Z ( g : g : · · · : g m +1 | t : · · · : t m : t m +1 ) = ( − m +1 Z ( g m +1 : · · · : g |− t m : · · · : − t : − t m +1 ) , for m (cid:62) . The following reformulation of the above dihedral symmetry relations is usefulin our later arguments. Remark 3.3. The transformation (3.4) allows us to rewrite the above dihedralsymmetry relations as follows: the cyclic symmetry relation (3.9) Z ∼ ( g , g , . . . , g m +1 | t : t : · · · : t m +1 ) = Z ∼ ( g , . . . , g m +1 , g | t : · · · : t m +1 : t ) , ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 31 the inversion relation (3.10) Z ∼ ( g , g , . . . , g m +1 | t : t : · · · : t m +1 ) = Z ∼ ( g − , g − , . . . , g − m +1 |− t : − t : · · · : − t m +1 ) , the reflection relation (3.11) Z ∼ ( g , . . . , g m , g m +1 | t : · · · : t m : t m +1 ) = ( − m +1 Z ∼ ( g − m , . . . , g − , g − m +1 |− t m : · · · : − t : − t m +1 ) with g g · · · g m +1 = 1 . Definition 3.4. Goncharov [G01a, §4.5 and §5.2] also introduced a related Liealgebra D (cid:48) (Γ) •• = ⊕ w,m D (cid:48) (Γ) w,m which consists of the collections Z ∼ satisfying the shuffle product (3.6), the cyclicsymmetry relation (3.2) and the additional condition (3.8) (cf. [G01a, Proposition4.6]). For our purpose, we consider its Q -linear subspace D (Γ) •• = ⊕ w,m D (Γ) w,m which consists of elements in D (cid:48) (Γ) •• satisfying the double shuffle relations (a).By Theorem 3.2, we have D (cid:48) (Γ) •• ⊃ D (Γ) •• ⊃ D (Γ) •• . Mould theoretic reformulation. We explain a reformulation of D (Γ) •• and D (Γ) •• in terms of moulds in Theorem 3.5 and Corollary 3.6 respectively.Let Z ∼ be a collection (3.1) of translation invariant elements. We associate amould MZ ∼ = ( M i Z ∼ ) i ∈ Z (cid:62) ∈ M ( F ; Γ) by M i Z ∼ = 0 for i (cid:54) = m and M m Z ∼ ( u ,...,u m g , ··· ,g m ) = ˜ Z ( g : · · · : g m : 1 | u , . . . , u m +1 ) . The following is a generalization of the results in [M], [Sch15] which treat thecase of Γ = { e } . Theorem 3.5. The map sending M : Z ∼ ∈ D (cid:48) (Γ) •• (cid:55)→ MZ ∼ ∈ ARI(Γ) forms a Liealgebra homomorphism and it induces an isomorphism between (3.12) Fil D D (Γ) •• (cid:39) Fil D ARI(Γ) fin , polal / al . Here the left hand side means the depth>1-part of D (Γ) •• and the right hand sidemeans the finite polynomial-valued part of the subset of ARI(Γ) al / al (cf. Definition1.20) consisting of M with depth>1.Proof. It is immediate that that (3.6) is equivalent to the condition for MZ ∼ beingin ARI(Γ) al . By (3.2), we have swap( M m Z ∼ ) (cid:0) g ,...,g m v ,...,v m (cid:1) = ˜ Z ( g · · · g m : g · · · g m − : · · · : g : 1 | v m , v m − − v m , . . . , v − v , v − v , − v )= Z ∼ ( g − m , g − m − , . . . , g − , g · · · g m | v m : · · · : v : v : 0) . Thus we see that (3.5) is equivalent to the condition for swap( MZ ∼ ) being in ARI(Γ) al . Therefore our map forms a Q -linear isomorphism (3.12) by Theorem3.2.Since MZ ∼ is in ARI(Γ) fin , polal when Z ∼ ∈ D (cid:48) (Γ) •• , by Proposition 2.8 there is an h ∈ L with depth m such that ma( h ) = MZ ∼ , that is, ma mh ( u ,...,u m g , ··· ,g m ) = ˜ Z ( g : · · · : g m : 1 | u , . . . , u m +1 ) with u + · · · + u m + u m +1 = 0 . By (2.3) and (3.3), we have vimo mh ( u ,...,u m g , ··· ,g m ) = Z ( g : · · · : g m : 1 | u : · · · : u m : u ) . In [G01a, Theorem 5.2], D (cid:48) (Γ) •• is realized as a Lie subalgebra of sder Γ underthe morphism(3.13) ξ (cid:48) Γ : D (cid:48) (Γ) •• → gr D sder Γ sending each Z ∼ = { Z ∼ ( g , . . . , g m +1 | t : · · · : t m +1 ) } g ,...,g m +1 ∈ Γ ∈ D w,m (Γ) to theresidue class of D { σ ( F ) } σ ,G ( F ) ∈ sder Γ in gr m D sder Γ with F = | Γ | − (cid:88) n ,...,nm +1 > g ,...,gm +1 ∈ Γ I n ,...,n m +1 ( g : · · · : g m +1 ) X n − Y g − g · · · X n m − Y g − g m +1 X n m +1 − = (cid:88) n ,...,nm> g ,...,gm ∈ Γ I n ,...,n m (1 : g : · · · : g m ) X n − Y g · · · X n m − − Y g m X n m − ∈ L w,m when the associated element Z = { Z ( g : · · · : g m +1 | t : · · · : t m +1 ) } g ,...,g m +1 ∈ Γ givenby (3.4) is expressed as Z ( g : · · · : g m +1 | t : · · · : t m +1 ) := (cid:88) n ,...,n m +1 > I n ,...,n m +1 ( g : · · · : g m +1 ) t n − · · · t n m +1 − m +1 . By combining the Lie algebra homomorphisms ξ (cid:48) Γ with (2.7) and (2.9), we obtain ma ◦ res ◦ ξ (cid:48) Γ ( Z ∼ ) = ma( F ) . By definition, we have vimo mF ( u ,...,u m g − ,...,g − m ) = (cid:88) n ,...,n m +1 > I n ,...,n m (1 : g : · · · : g m ) u n − · · · u n m − m = Z (1 : g : · · · : g m | u : · · · : u m ) . By the cyclic symmetry relation (3.2), = Z ( g : · · · : g m : 1 | u : · · · : u m : u ) = vimo mh ( u ,...,u m g ,...,g m ) . Therefore we have F ( x, ( y σ ) σ ) = h ( x, ( y σ − ) σ ) . So M ( Z ∼ ) = ma( h ) = ι ◦ ma ◦ res ◦ ξ (cid:48) Γ ( Z ∼ ) where ι is the map defined by ( ιM ) m ( u ,...,u m g ,...,g m ) = M m ( u ,...,u m g − ,...,g − m ) which forms a Liealgebra homomorphism. Since ι , ma in (2.9), res in (2.7) and ξ (cid:48) Γ in (3.13) are allLie algebra homomorphisms, we see that M is so. Whence we get our claims. (cid:3) ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 33 Since ARI(Γ) al / al , and hence the right hand side of (3.12), forms a Lie algebraby Proposition 1.21, we learn that Fil D D (Γ) •• forms a Lie subalgebra of D (cid:48) (Γ) •• . Corollary 3.6. The map M in Theorem 3.5 induces a Lie algebra isomorphismbetween (3.14) Fil D D (Γ) •• (cid:39) Fil D ARID(Γ) fin , polal / al . Here the left hand side means the depth>1-part of D (Γ) •• and the right hand sidemeans the finite polynomial-valued part of the subset of ARID(Γ) al / al (cf. Definition1.30) consisting of M with depth>1.Proof. Since the distribution relation (3.7) is equivalent to Z ∼ ( g , . . . , g m +1 | t : · · · : t m +1 ) = (cid:88) h Ni = g i Z ∼ ( h , . . . , h m +1 | N t : · · · : N t m +1 ) , which corresponds to i N ( MZ ∼ ) = m N ( MZ ∼ ) , that is, MZ ∼ ∈ ARID(Γ) . Since ARID(Γ) al / al , and hence the right hand side of (3.14), forms a Lie algebra byCorollary 1.32, and Fil D D (Γ) •• forms a Lie subalgebra of D (Γ) •• by [G01a, Theo-rem 5.2], our claim follows. (cid:3) Embedding. In this subsection, we construct an embedding Fil D ARI(Γ) al / al (cid:44) → ARI(Γ) push / pusnu in Theorem 3.13. As a corollary, we get an embedding Fil D D (Γ) •• (cid:44) → lkrv (Γ) •• in Corollary 3.14.The following generalizes [Sch15, Lemma 2.5.3]. Lemma 3.7. Any mould M ∈ ARI(Γ) al is mantar -invariant (cf. Notation 1.6),namely, for m (cid:62) and σ , . . . , σ m ∈ Γ , we have (3.15) M m (cid:0) x , ..., x m σ , ..., σ m (cid:1) = ( − m − M m (cid:0) x m , ..., x σ m , ..., σ (cid:1) . Proof. For simplicity, we denote ω i := (cid:0) x i σ i (cid:1) . By using alternality of M , we have m − (cid:88) i =1 ( − i − (cid:88) α ∈ X • Z Sh Ç ( ω i , . . . , ω ); ( ω i +1 , . . . , ω m ) α å M m ( α ) = 0 . Here, we calculate the left hand side as follows: m − (cid:88) i =1 ( − i − (cid:88) α ∈ X • Z Sh Ç ( ω i , . . . , ω ); ( ω i +1 , . . . , ω m ) α å M m ( α ) = m − (cid:88) i =1 ( − i − (cid:88) α ∈ X • Z Sh Ç ( ω i − , . . . , ω ); ( ω i +1 , . . . , ω m ) α å M m ( ω i , α )+ (cid:88) α ∈ X • Z Sh Ç ( ω i , . . . , ω ); ( ω i +2 , . . . , ω m ) α å M m ( ω i +1 , α ) = m − (cid:88) i =1 ( − i − (cid:88) α ∈ X • Z Sh Ç ( ω i − , . . . , ω ); ( ω i +1 , . . . , ω m ) α å M m ( ω i , α ) − m (cid:88) i =2 ( − i − (cid:88) α ∈ X • Z Sh Ç ( ω i − , . . . , ω ); ( ω i +1 , . . . , ω m ) α å M m ( ω i , α )= M m ( ω , ω , . . . , ω m ) − ( − m − M m ( ω m , ω m − , . . . , ω ) . So we obtain (3.15). (cid:3) We define the parallel translation map t ∼ : ARI(Γ) → ARI(Γ) by t ∼ ( M ) m ( y m ) := ® M m ( y m ) ( m = 0 , ,M m − Ä σ , . . . , σ m y − y , . . . , y m − y ä ( m (cid:62) , for M ∈ ARI(Γ) . For our simplicity, we put u ∼ := t ∼ ◦ swap . Lemma 3.8. Let M ∈ ARI(Γ) al / al and m (cid:62) and σ , . . . , σ m ∈ Γ with σ · · · σ m =1 . Then we have (3.16) u ∼ ( M ) m (cid:0) σ , ..., σ m x , ..., x m (cid:1) = u ∼ ( M ) m (cid:16) σ − , ..., σ − m , σ − − x , ..., − x m , − x (cid:17) . Proof. We have u ∼ ( M ) m (cid:0) σ , ..., σ m x , ..., x m (cid:1) =swap( M ) m − Ä σ , σ , ..., σ m − , σ m x − x , x − x , ..., x m − − x , x m − x ä = M m − Ä x m − x , x m − − x m , ..., x − x σ ··· σ m , σ ··· σ m − , ..., σ ä . By using (3.15), we get =( − m − M m − Ä x − x , ..., x m − − x m , x m − x σ , ..., σ ··· σ m − , σ ··· σ m ä =( − m − swap( M ) m − (cid:16) σ ··· σ m , σ − m , ..., σ − x − x , x − x m , ..., x − x (cid:17) . By (3.15), σ · · · σ m = 1 and M ∈ ARI(Γ) al / al , we calculate =swap( M ) m − (cid:16) σ − , ..., σ − m , σ − x − x , ..., x − x m , x − x (cid:17) = u ∼ ( M ) m (cid:16) σ − , σ − , ..., σ − m , σ − − x , − x , ..., − x m , − x (cid:17) . ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 35 Therefore, we obtain the claim. (cid:3) 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 ). Lemma 3.9. Let m (cid:62) and M ∈ ARI(Γ) al / al . Then, for σ , . . . , σ m ∈ Γ with σ · · · σ m = 1 , we have the following: u ∼ ( M ) m (cid:0) σ , σ , ..., σ m x , x , ..., x m (cid:1) = u ∼ ( M ) m (cid:0) σ , ..., σ m , σ x , ..., x m , x (cid:1) , (3.17) u ∼ ( M ) m (cid:0) σ , σ , ..., σ m x , x , ..., x m (cid:1) = u ∼ ( M ) m (cid:16) σ − , σ − , ..., σ − m − x , − x , ..., − x m (cid:17) , (3.18) u ∼ ( M ) m Ä σ , ..., σ m − , σ m x , ..., x m − , x m ä = ( − m u ∼ ( M ) m Å σ − m − , ..., σ − , σ − m − x m − , ..., − x , − x m ã . (3.19) Proof. It is easy to get (3.18) from (3.16) and (3.17) and to get (3.19) from (3.15),(3.17) and (3.18). So it is enough to prove (3.17).Firstly, we have (cid:88) α ∈ Y • Z Sh Ç (cid:0) σ x (cid:1) ; (cid:0) σ , ..., σ m x , ..., x m (cid:1) α å u ∼ ( M ) m ( α )= u ∼ ( M ) m (cid:0) σ , σ , ..., σ m x , x , ..., x m (cid:1) + (cid:88) α ∈ Y • Z Sh Ç (cid:0) σ x (cid:1) ; (cid:0) σ , ..., σ m x , ..., x m (cid:1) α å u ∼ ( M ) m ( (cid:0) σ x (cid:1) , α )= u ∼ ( M ) m (cid:0) σ , ..., σ m x , ..., x m (cid:1) + (cid:88) α ∈ Y • Z Sh Ç (cid:0) σ x − x (cid:1) ; (cid:0) σ , ..., σ m x − x , ..., x m − x (cid:1) α å swap( M ) m − ( α ) . So by using alternality of swap( M ) , we obtain(3.20) (cid:88) α ∈ Y • Z Sh Ç (cid:0) σ x (cid:1) ; (cid:0) σ , ..., σ m x , ..., x m (cid:1) α å u ∼ ( M ) m ( α ) = u ∼ ( M ) m (cid:0) σ , ..., σ m x , ..., x m (cid:1) . Secondly, we have (cid:88) α ∈ Y • Z Sh Ç (cid:0) σ x (cid:1) ; (cid:0) σ , ..., σ m x , ..., x m (cid:1) α å u ∼ ( M ) m ( α )= u ∼ ( M ) m (cid:0) σ , ..., σ m , σ x , ..., x m , x (cid:1) + (cid:88) α ∈ Y • Z Sh Ç (cid:0) σ x (cid:1) ; (cid:0) σ , ..., σ m − x , ..., x m − (cid:1) α å u ∼ ( M ) m ( α, (cid:0) σ m x m (cid:1) ) . By applying Lemma 3.8 to the second term, we get = u ∼ ( M ) m (cid:0) σ , ..., σ m , σ x , ..., x m , x (cid:1) + (cid:88) α ∈ Y • Z Sh Ç (cid:0) σ x (cid:1) ; (cid:0) σ , ..., σ m − x , ..., x m − (cid:1) α å u ∼ ( M ) m ( (cid:16) σ − m − x m (cid:17) , α − )= u ∼ ( M ) m (cid:0) σ , ..., σ m , σ x , ..., x m , x (cid:1) + (cid:88) α ∈ Y • Z Sh Ç (cid:0) σ x − x m (cid:1) ; Ä σ , ..., σ m − x − x m , ..., x m − − x m ä α å swap( M ) m − ( α − ) . Here, the symbol α − is (cid:16) τ − ,...,τ − m − u ,..., − u m (cid:17) for α = (cid:0) τ ,...,τ m u ,...,u m (cid:1) and for (cid:0) τ u (cid:1) , . . . , (cid:0) τ m u m (cid:1) ∈ Y Z .So by using alternality of swap( M ) , we obtain(3.21) (cid:88) α ∈ Y • Z Sh Ç (cid:0) σ x (cid:1) ; (cid:0) σ , ..., σ m x , ..., x m (cid:1) α å u ∼ ( M ) m ( α ) = u ∼ ( M ) m (cid:0) σ , ..., σ m , σ x , ..., x m , x (cid:1) . Therefore, by combining (3.20) and (3.21), we obtain (3.17). (cid:3) Lemma 3.10. For any mould M ∈ ARI(Γ) al / al , its swap( M ) is neg-invariant andmantar-invariant (cf. Notation 1.6) namely we have swap( M ) m (cid:0) σ , ..., σ m x , ..., x m (cid:1) = swap( M ) m (cid:16) σ − , ..., σ − m − x , ..., − x m (cid:17) , (3.22) swap( M ) m (cid:0) σ , ..., σ m x , ..., x m (cid:1) = ( − m − swap( M ) m (cid:16) σ − m , ..., σ − − x m , ..., − x (cid:17) , (3.23) for m (cid:62) .Proof. For m (cid:62) , the first equation follows from (3.18) and the second one followsfrom (3.17) and (3.19). For m = 0 , , they follow from the definition of ARI(Γ) al / al . (cid:3) The following can be also found in [Sch15, Lemma 2.5.5] but we give a differentproof below. Proposition 3.11. Any mould M ∈ ARI(Γ) al / al is push-invariant (1.4) .Proof. For m = 0 , , it is obvious because we have M (cid:0) x σ (cid:1) = M (cid:16) − x σ − (cid:17) . Assume m (cid:62) . By using (3.17), we have u ∼ ( M ) m +1 Ä τ , τ , ..., τ m +1 y , y , ..., y m +1 ä = u ∼ ( M ) m +1 Ä τ , ..., τ m +1 , τ y , ..., y m +1 , y ä , for (cid:0) τ y (cid:1) , . . . , (cid:0) τ m +1 y m +1 (cid:1) ∈ Y Z with τ · · · τ m +1 = 1 . We calculate the left hand side u ∼ ( M ) m +1 Ä τ , τ , ..., τ m +1 y , y , ..., y m +1 ä = swap( M ) m Ä τ , ..., τ m +1 y − y , ..., y m +1 − y ä = M m Ä y m +1 − y , y m − y m +1 , . . . , y − y τ · · · τ m +1 , τ · · · τ m , . . . , τ ä . Similarly, we calculate the right hand side u ∼ ( M ) m +1 Ä τ , ..., τ m +1 , τ y , ..., y m +1 , y ä = M m Ä y − y , y m +1 − y , y m − y m +1 , . . . , y − y τ · · · τ m +1 τ , τ · · · τ m +1 , τ · · · τ m , . . . , τ ä . By substituting y := 0 and y i := x + · · · + x m +2 − i ( (cid:54) i (cid:54) m + 1 ) and τ := σ − and τ := σ m and τ j := σ m +2 − j σ − m +3 − j ( (cid:54) j (cid:54) m + 1 ) for any σ , . . . , σ m ∈ Γ ,we have M m (cid:0) x , x , ..., x m σ , σ , ..., σ m (cid:1) = M m Ä − x − · · · − x m , x , . . . , x m − σ − m , σ σ − m , . . . , σ m − σ − m ä = push( M ) m (cid:0) x , x , ..., x m σ , σ , ..., σ m (cid:1) . Whence we obtain the claim. (cid:3) The following generalizes [RS, Theorem 14]. Proposition 3.12. For any mould M ∈ Fil D ARI(Γ) al / al , its swap swap( M ) ispus-neutral (1.8) , that is, for all m (cid:62) and σ , . . . , σ m ∈ Γ , (cid:88) i ∈ Z /m Z pus i ◦ swap( M ) m (cid:0) σ ,...,σ m x ,...,x m (cid:1) = 0 . ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 37 Proof. For m = 0 , , it is obvious because we have M ( x ) = 0 . Assume m (cid:62) .Let (cid:0) τ u (cid:1) , (cid:0) τ u (cid:1) , . . . , (cid:0) τ m u m (cid:1) ∈ Y Z with τ τ · · · τ m = 1 . By using alternality of swap( M ) ,we have (cid:88) α ∈ Y • Z Sh ÇÄ τ , . . . , τ m u − u , . . . , u m − u ä ; Ä τ u − u ä α å swap( M ) m ( α ) = 0 . On the other hand, we calculate (cid:88) α ∈ Y • Z Sh ÇÄ τ , . . . , τ m u − u , . . . , u m − u ä ; Ä τ u − u ä α å swap( M ) m ( α )= u ∼ ( M ) m +1 (cid:0) τ , τ , ..., τ m , τ u , u , ..., u m , u (cid:1) + u ∼ ( M ) m +1 Ä τ , τ , ..., τ m − , τ , τ m u , u , ..., u m − , u , u m ä + · · · + u ∼ ( M ) m +1 (cid:0) τ , τ , τ , τ , ..., τ m u , u , u , u , ..., u m (cid:1) + u ∼ ( M ) m +1 (cid:0) τ , τ , τ , ..., τ m u , u , u , ..., u m (cid:1) . Repeated applications of (3.17) yield = u ∼ ( M ) m +1 (cid:0) τ , τ , τ , ..., τ m u , u , u , ..., u m (cid:1) + u ∼ ( M ) m +1 Ä τ , τ m , τ , τ , ..., τ m − u , u m , u , u , ..., u m − ä + · · · + u ∼ ( M ) m +1 (cid:0) τ , τ , ..., τ m , τ , τ u , u , ..., u m , u , u (cid:1) + u ∼ ( M ) m +1 (cid:0) τ , τ , ..., τ m , τ u , u , ..., u m , u (cid:1) = (cid:88) i ∈ Z /m Z u ∼ ( M ) m +1 Ä τ , τ i +1 , τ i +2 , ..., τ i + m u , u i +1 , u i +2 , ..., u i + m ä . Therefore, by substituting u = 0 and u i = x i ( (cid:54) i (cid:54) m ) and τ = ( σ · · · σ m ) − and τ i = σ i ( (cid:54) i (cid:54) m ), we get the claim. (cid:3) Theorem 3.13. There is an embedding Fil D ARI(Γ) al / al (cid:44) → ARI(Γ) push / pusnu of graded Lie algebras.Proof. Let M ∈ Fil D ARI(Γ) al / al . Then we have M ∈ ARI(Γ) al / al . So by Proposi-tion 3.11 and Proposition 3.12, M is push -invariant and swap( M ) is pus -neutral.Therefore, we obtain M ∈ ARI(Γ) push / pusnu . To see that it is a Lie algebra homo-morphism is immediate. (cid:3) As a corollary, by taking an intersection with ARI(Γ) fin , polal in the embeddingof the above theorem, we obtain the following inclusion which generalizes [RS,Theorem 3]. Corollary 3.14. There is an embedding Fil D D (Γ) •• (cid:44) → lkrv (Γ) •• of bigraded Lie algebras.Proof. It follows from Theorem 2.22, (3.12) and Theorem 3.13. (cid:3) By imposing the distribution relation, we also obtain the following. Corollary 3.15. There is an embedding Fil D D (Γ) •• (cid:44) → lkrvd (Γ) •• of bigraded Lie algebras.Proof. It follows from Corollary 2.26, Corollary 3.6 and Theorem 3.13. (cid:3) 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, weobtain the commutative diagram (3.24) of Lie algebras:(3.24) Fil D D (Γ) •• (cid:39) (cid:47) (cid:47) (cid:127) (cid:95) (cid:15) (cid:15) Fil D ARI(Γ) fin , polal / al (cid:127) (cid:95) (cid:15) (cid:15) lkrv (Γ) •• (cid:39) (cid:47) (cid:47) ARI(Γ) push / pusnu ∩ ARI(Γ) fin , polal Appendix A. 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 1.12, 1.13 and 1.21, which are requiredin this paper.A.1. Proof of Proposition 1.12. We give a proof Proposition 1.12 which claimsthat 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 simplecomputations. Lemma A.1. For α, α , α , β, β , β ∈ X • Z , we have α α (cid:100) β = α (cid:100) ( α (cid:100) β ) , β (cid:101) α α = ( β (cid:101) α ) (cid:101) α , α (cid:98) ( β β ) = α (cid:98) β α (cid:98) β , ( β β ) (cid:99) α = β (cid:99) α β (cid:99) α . Especially, if β (cid:54) = ∅ , we have α (cid:100) ( β β ) = ( α (cid:100) β ) β , ( β β ) (cid:101) α = β ( β (cid:101) α ) , and if α (cid:54) = ∅ , we have α α (cid:98) β = α (cid:98) β, β (cid:99) α α = β (cid:99) α . Proof. Let α = (cid:0) a ,...,a l σ ,...,σ l (cid:1) , α = (cid:0) a l +1 ,...,a l + m σ l +1 ,...,σ l + m (cid:1) , β = (cid:0) b ,...,b n τ ,...,τ n (cid:1) . By α α = (cid:0) a ,...,a l + m σ ,...,σ l + m (cid:1) ,we have α α (cid:100) β = (cid:0) a + ··· + a l + m + b ,b ,...,b n τ ,τ ,...,τ n (cid:1) = α (cid:100) (cid:0) a l +1 + ··· + a l + m + b ,b ,...,b n τ ,τ ,...,τ n (cid:1) = α (cid:100) ( α (cid:100) β ) . All the other cases can shown similarly. (cid:3) Lemma A.2. Let a, b, c, d, e, f ∈ X • Z . (1) If we have a b (cid:100) c = def , one of the following holds: (I) : There exists a , a , a ∈ X • Z such that a = a a a and ( d, e, f ) = ( a , a , a b (cid:100) c ) , (II) : There exists a , a , c , c ∈ X • Z such that a = a a and c = c c and ( d, e, f ) = ( a , a b (cid:100) c , c ) and c (cid:54) = ∅ , (III) : There exists c , c , c ∈ X • Z such that c = c c c and ( d, e, f ) = ( a b (cid:100) c , c , c ) and c (cid:54) = ∅ . ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 39 (2) If we have a (cid:101) b c = def , one of the followings holds: (I) : There exists a , a , a ∈ X • Z such that a = a a a and ( d, e, f ) = ( a , a , a (cid:101) b c ) and a (cid:54) = ∅ , (II) : There exists a , a , c , c ∈ X • Z such that a = a a and c = c c and ( d, e, f ) = ( a , a (cid:101) b c , c ) and a (cid:54) = ∅ , (III) : There exists c , c , c ∈ X • Z such that c = c c c and ( d, e, f ) = ( a (cid:101) b c , c , c ) .Proof. We present a proof for (1). When a b (cid:100) c = def , the following depict all thepossible cases of the positions of a , b (cid:100) c , d , e and f . a (cid:122) (cid:125)(cid:124) (cid:123) ( (cid:124)(cid:123)(cid:122)(cid:125) d | (cid:124)(cid:123)(cid:122)(cid:125) e | ) b (cid:100) c (cid:122) (cid:125)(cid:124) (cid:123) ( ) (cid:124) (cid:123)(cid:122) (cid:125) f , a (cid:122) (cid:125)(cid:124) (cid:123) ( (cid:124)(cid:123)(cid:122)(cid:125) d | ) b (cid:100) c (cid:122) (cid:125)(cid:124) (cid:123) ( | (cid:124)(cid:123)(cid:122)(cid:125) f ) (cid:124) (cid:123)(cid:122) (cid:125) e , a (cid:122) (cid:125)(cid:124) (cid:123) ( ) b (cid:100) c (cid:122) (cid:125)(cid:124) (cid:123) ( | (cid:124)(cid:123)(cid:122)(cid:125) e | (cid:124)(cid:123)(cid:122)(cid:125) f ) . (cid:124) (cid:123)(cid:122) (cid:125) d The first, the second and the third cases correspond (I), (II) and (III) in (1) respec-tively . The claim for (2) can be proved in the same way. (cid:3) Lemma A.3. For a, b, c ∈ X • Z , the following hold: (1) (commutativity) a (cid:100) ( b (cid:100) c ) = b (cid:100) ( a (cid:100) c ) , ( c (cid:101) a ) (cid:101) b = ( c (cid:101) b ) (cid:101) a , a (cid:98) ( b (cid:98) c ) = b (cid:98) ( a (cid:98) c ) , ( c (cid:99) a ) (cid:99) b = ( c (cid:99) b ) (cid:99) a , a (cid:100) ( c (cid:101) b ) = ( a (cid:100) c ) (cid:101) b , a (cid:100) ( b (cid:98) c ) = b (cid:98) ( a (cid:100) c ) , a (cid:100) ( c (cid:99) b ) = ( a (cid:100) c ) (cid:99) b , ( b (cid:98) c ) (cid:101) a = b (cid:98) ( c (cid:101) a ) , ( c (cid:99) b ) (cid:101) a = ( c (cid:101) a ) (cid:99) b , a (cid:98) ( c (cid:99) b ) = ( a (cid:98) c ) (cid:99) b . (2) (composition) ( b (cid:101) a ) (cid:100) c = ( a (cid:100) b ) (cid:100) c = ab (cid:100) c, ( b (cid:99) a ) (cid:98) ( c (cid:99) a ) = ( a (cid:98) b ) (cid:98) ( a (cid:98) c ) = b (cid:98) c,c (cid:101) ( b (cid:101) a ) = c (cid:101) ( a (cid:100) b ) = c (cid:101) ab , ( c (cid:99) a ) (cid:99) ( b (cid:99) a ) = ( a (cid:98) c ) (cid:99) ( a (cid:98) b ) = c (cid:99) b . (3) (independence) ( b (cid:99) a ) (cid:100) c = ( a (cid:98) b ) (cid:100) c = b (cid:100) c, ( b (cid:101) a ) (cid:98) c = ( a (cid:100) b ) (cid:98) c = b (cid:98) c,c (cid:101) ( b (cid:99) a ) = c (cid:101) ( a (cid:98) b ) = c (cid:101) b , c (cid:99) ( b (cid:101) a ) = c (cid:99) ( a (cid:100) b ) = c (cid:99) b . Proof. Let a = (cid:0) a ,...,a l σ ,...,σ l (cid:1) , b = (cid:0) b ,...,b m τ ,...,τ m (cid:1) , c = (cid:0) c ,...,c n µ ,...,µ n (cid:1) . We give proofs for specificcases because all the other cases can be proved in a similar way.(1). We calculate a (cid:100) ( b (cid:100) c ) = a (cid:100) (cid:0) b + ··· + b m + c ,c ,...,c n µ ,µ ,...,µ n (cid:1) = (cid:0) a + ··· + a l + b + ··· + b m + c ,c ,...,c n µ ,µ ,...,µ n (cid:1) = b (cid:100) (cid:0) a + ··· + a l + c ,c ,...,c n µ ,µ ,...,µ n (cid:1) = b (cid:100) ( a (cid:100) c ) . (2). By b (cid:101) a = Ä b ,...,b m − ,b m + a ,...,a l τ ,...,τ m − ,τ m ä and ab = (cid:0) a ,...,a l ,b ,...,b m σ ,...,σ l ,τ ,...,τ m (cid:1) , we get ( b (cid:101) a ) (cid:100) c = (cid:0) a + ··· + a l + b + ··· + b m + c ,c ,...,c n µ ,µ ,...,µ n (cid:1) = ab (cid:100) c. On the other hand, by b (cid:99) a = (cid:16) b ,..., b m τ σ − ,...,τ m σ − (cid:17) and c (cid:99) a = (cid:16) c ,..., c n µ σ − ,...,µ n σ − (cid:17) , we get ( b (cid:99) a ) (cid:98) ( c (cid:99) a ) = (cid:16) c ,..., c n µ τ − m ,...,µ n τ − m (cid:17) = b (cid:98) c. Note that, in order to decompose b (cid:100) c , we need the condition c (cid:54) = ∅ for the second and thirdcases. (3). By the above expression of b (cid:99) a , we get ( b (cid:99) a ) (cid:100) c = (cid:0) b + ··· + b m + c ,c ,...,c n µ ,µ ,...,µ n (cid:1) = b (cid:100) c. (cid:3) The following formula is essential to prove Proposition 1.12. It is stated in[Sch15, (2.2.10)] (it looks that there is an error on the signature) without a proof. Proposition A.4. For any A, B ∈ ARI(Γ) , we have arit u ( B ) ◦ arit u ( A ) − arit u ( A ) ◦ arit u ( B ) = arit u (ari u ( A, B )) . Proof. Let m (cid:62) . Then we have (arit u ( B ) ◦ arit u ( A ) − arit u ( A ) ◦ arit u ( B ))( C )( x m )= arit u ( B )(arit u ( A )( C ))( x m ) − arit u ( A )(arit u ( B )( C ))( x m )= (cid:88) x m = abcb,c (cid:54) = ∅ arit u ( A )( C )( a b (cid:100) c ) B ( b (cid:99) c ) − (cid:88) x m = abca,b (cid:54) = ∅ arit u ( A )( C )( a (cid:101) b c ) B ( a (cid:98) b ) − (cid:88) x m = abcb,c (cid:54) = ∅ arit u ( B )( C )( a b (cid:100) c ) A ( b (cid:99) c ) + (cid:88) x m = abca,b (cid:54) = ∅ arit u ( B )( C )( a (cid:101) b c ) A ( a (cid:98) b )= (cid:88) x m = abcb,c (cid:54) = ∅ (cid:88) a b (cid:100) c = defe,f (cid:54) = ∅ C ( d e (cid:100) f ) A ( e (cid:99) f ) − (cid:88) a b (cid:100) c = defd,e (cid:54) = ∅ C ( d (cid:101) e f ) A ( d (cid:98) e ) B ( b (cid:99) c ) − (cid:88) x m = abca,b (cid:54) = ∅ (cid:88) a (cid:101) b c = defe,f (cid:54) = ∅ C ( d e (cid:100) f ) A ( e (cid:99) f ) − (cid:88) a (cid:101) b c = defd,e (cid:54) = ∅ C ( d (cid:101) e f ) A ( d (cid:98) e ) B ( a (cid:98) b ) − (cid:88) x m = abcb,c (cid:54) = ∅ (cid:88) a b (cid:100) c = defe,f (cid:54) = ∅ C ( d e (cid:100) f ) B ( e (cid:99) f ) − (cid:88) a b (cid:100) c = defd,e (cid:54) = ∅ C ( d (cid:101) e f ) B ( d (cid:98) e ) A ( b (cid:99) c )+ (cid:88) x m = abca,b (cid:54) = ∅ (cid:88) a (cid:101) b c = defe,f (cid:54) = ∅ C ( d e (cid:100) f ) B ( e (cid:99) f ) − (cid:88) a (cid:101) b c = defd,e (cid:54) = ∅ C ( d (cid:101) e f ) B ( d (cid:98) e ) A ( a (cid:98) b ) . ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 41 By using Lemma A.2, we have = (cid:88) x m = abcb,c (cid:54) = ∅ (cid:88) a = a a a a (cid:54) = ∅ C ( a a (cid:100) ( a b (cid:100) c )) A ( a (cid:99) a b (cid:100) c ) + (cid:88) a = a a c = c c c ,c (cid:54) = ∅ C ( a a b (cid:100) c (cid:100) c ) A (( a b (cid:100) c ) (cid:99) c ) + (cid:88) c = c c c c ,c ,c (cid:54) = ∅ C ( a b (cid:100) c c (cid:100) c ) A ( c (cid:99) c ) − (cid:88) c = c c c c ,c (cid:54) = ∅ C (( a b (cid:100) c ) (cid:101) c c ) A ( a b (cid:100) c (cid:98) c ) − (cid:88) a = a a c = c c a ,c (cid:54) = ∅ C ( a (cid:101) a b (cid:100) c c ) A ( a (cid:98) ( a b (cid:100) c )) − (cid:88) a = a a a a ,a (cid:54) = ∅ C ( a (cid:101) a a b (cid:100) c ) A ( a (cid:98) a ) B ( b (cid:99) c ) − (cid:88) x m = abca,b (cid:54) = ∅ (cid:88) a = a a a a ,a (cid:54) = ∅ C ( a a (cid:100) ( a (cid:101) b c )) A ( a (cid:99) a (cid:101) b c ) + (cid:88) a = a a c = c c a ,c (cid:54) = ∅ C ( a a (cid:101) b c (cid:100) c ) A (( a (cid:101) b c ) (cid:99) c ) + (cid:88) c = c c c c ,c (cid:54) = ∅ C ( a (cid:101) b c c (cid:100) c ) A ( c (cid:99) c ) − (cid:88) c = c c c c (cid:54) = ∅ C (( a (cid:101) b c ) (cid:101) c c ) A ( a (cid:101) b c (cid:98) c ) − (cid:88) a = a a c = c c a ,a (cid:54) = ∅ C ( a (cid:101) a (cid:101) b c c ) A ( a (cid:98) ( a (cid:101) b c )) − (cid:88) a = a a a a ,a ,a (cid:54) = ∅ C ( a (cid:101) a a (cid:101) b c ) A ( a (cid:98) a ) B ( a (cid:98) b ) − (cid:88) x m = abcb,c (cid:54) = ∅ (cid:88) a = a a a a (cid:54) = ∅ C ( a a (cid:100) ( a b (cid:100) c )) B ( a (cid:99) a b (cid:100) c ) + (cid:88) a = a a c = c c c ,c (cid:54) = ∅ C ( a a b (cid:100) c (cid:100) c ) B (( a b (cid:100) c ) (cid:99) c ) + (cid:88) c = c c c c ,c ,c (cid:54) = ∅ C ( a b (cid:100) c c (cid:100) c ) B ( c (cid:99) c ) − (cid:88) c = c c c c ,c (cid:54) = ∅ C (( a b (cid:100) c ) (cid:101) c c ) B ( a b (cid:100) c (cid:98) c ) − (cid:88) a = a a c = c c a ,c (cid:54) = ∅ C ( a (cid:101) a b (cid:100) c c ) B ( a (cid:98) ( a b (cid:100) c )) − (cid:88) a = a a a a ,a (cid:54) = ∅ C ( a (cid:101) a a b (cid:100) c ) B ( a (cid:98) a ) A ( b (cid:99) c )+ (cid:88) x m = abca,b (cid:54) = ∅ (cid:88) a = a a a a ,a (cid:54) = ∅ C ( a a (cid:100) ( a (cid:101) b c )) B ( a (cid:99) a (cid:101) b c ) + (cid:88) a = a a c = c c a ,c (cid:54) = ∅ C ( a a (cid:101) b c (cid:100) c ) B (( a (cid:101) b c ) (cid:99) c ) + (cid:88) c = c c c c ,c (cid:54) = ∅ C ( a (cid:101) b c c (cid:100) c ) B ( c (cid:99) c ) − (cid:88) c = c c c c (cid:54) = ∅ C (( a (cid:101) b c ) (cid:101) c c ) B ( a (cid:101) b c (cid:98) c ) − (cid:88) a = a a c = c c a ,a (cid:54) = ∅ C ( a (cid:101) a (cid:101) b c c ) B ( a (cid:98) ( a (cid:101) b c )) − (cid:88) a = a a a a ,a ,a (cid:54) = ∅ C ( a (cid:101) a a (cid:101) b c ) B ( a (cid:98) a ) A ( a (cid:98) b ) . By using Lemma A.1 and Lemma A.3.(2), (3) and changing variables, wecalculate = (cid:88) x m = abcdeb,d,e (cid:54) = ∅ C ( a b (cid:100) ( c d (cid:100) e )) A ( b (cid:99) ce ) B ( d (cid:99) e ) + (cid:88) x m = abcdec,d,e (cid:54) = ∅ C ( a bcd (cid:100) e ) A (( b c (cid:100) d ) (cid:99) e ) B ( c (cid:99) de ) + (cid:88) x m = abcdeb,c,d,e (cid:54) = ∅ C ( a b (cid:100) c d (cid:100) e ) A ( d (cid:99) e ) B ( b (cid:99) cde ) − (cid:88) x m = abcdeb,c,d (cid:54) = ∅ C ( a b (cid:100) c (cid:101) d e ) A ( ac (cid:98) d ) B ( b (cid:99) cde ) − (cid:88) x m = abcdea,c,d (cid:54) = ∅ C ( a (cid:101) bcd e ) A ( a (cid:98) ( b c (cid:100) d )) B ( c (cid:99) de ) − (cid:88) x m = abcdea,b,d,e (cid:54) = ∅ C ( a (cid:101) b c d (cid:100) e ) A ( a (cid:98) b ) B ( d (cid:99) e ) − (cid:88) x m = abcdeb,c,d (cid:54) = ∅ C ( a b (cid:100) c (cid:101) d e ) A ( b (cid:99) ce ) B ( abc (cid:98) d ) − (cid:88) x m = abcdeb,c,e (cid:54) = ∅ C ( a bcd (cid:100) e ) A (( b (cid:101) c d ) (cid:99) e ) B ( ab (cid:98) c ) − (cid:88) x m = abcdea,b,d,e (cid:54) = ∅ C ( a (cid:101) b c d (cid:100) e ) A ( d (cid:99) e ) B ( a (cid:98) b )+ (cid:88) x m = abcdea,b,d (cid:54) = ∅ C (( a (cid:101) b c ) (cid:101) d e ) A ( ac (cid:98) d ) B ( a (cid:98) b ) + (cid:88) x m = abcdea,b,c (cid:54) = ∅ C ( a (cid:101) bcd e ) A ( a (cid:98) ( b (cid:101) c d )) B ( ab (cid:98) c ) + (cid:88) x m = abcdea,b,c,d (cid:54) = ∅ C ( a (cid:101) b c (cid:101) d e ) A ( a (cid:98) b ) B ( abc (cid:98) d ) − (cid:88) x m = abcdeb,d,e (cid:54) = ∅ C ( a b (cid:100) ( c d (cid:100) e )) B ( b (cid:99) ce ) A ( d (cid:99) e ) − (cid:88) x m = abcdec,d,e (cid:54) = ∅ C ( a bcd (cid:100) e ) B (( b c (cid:100) d ) (cid:99) e ) A ( c (cid:99) de ) − (cid:88) x m = abcdeb,c,d,e (cid:54) = ∅ C ( a b (cid:100) c d (cid:100) e ) B ( d (cid:99) e ) A ( b (cid:99) cde )+ (cid:88) x m = abcdeb,c,d (cid:54) = ∅ C ( a b (cid:100) c (cid:101) d e ) B ( ac (cid:98) d ) A ( b (cid:99) cde ) + (cid:88) x m = abcdea,c,d (cid:54) = ∅ C ( a (cid:101) bcd e ) B ( a (cid:98) ( b c (cid:100) d )) A ( c (cid:99) de ) + (cid:88) x m = abcdea,b,d,e (cid:54) = ∅ C ( a (cid:101) b c d (cid:100) e ) B ( a (cid:98) b ) A ( d (cid:99) e )+ (cid:88) x m = abcdeb,c,d (cid:54) = ∅ C ( a b (cid:100) c (cid:101) d e ) B ( b (cid:99) ce ) A ( abc (cid:98) d ) + (cid:88) x m = abcdeb,c,e (cid:54) = ∅ C ( a bcd (cid:100) e ) B (( b (cid:101) c d ) (cid:99) e ) A ( ab (cid:98) c ) + (cid:88) x m = abcdea,b,d,e (cid:54) = ∅ C ( a (cid:101) b c d (cid:100) e ) B ( d (cid:99) e ) A ( a (cid:98) b ) − (cid:88) x m = abcdea,b,d (cid:54) = ∅ C (( a (cid:101) b c ) (cid:101) d e ) B ( ac (cid:98) d ) A ( a (cid:98) b ) − (cid:88) x m = abcdea,b,c (cid:54) = ∅ C ( a (cid:101) bcd e ) B ( a (cid:98) ( b (cid:101) c d )) A ( ab (cid:98) c ) − (cid:88) x m = abcdea,b,c,d (cid:54) = ∅ C ( a (cid:101) b c (cid:101) d e ) B ( a (cid:98) b ) A ( abc (cid:98) d ) . We apply Lemma A.1 to the 4th, 7th, 16th and 19th terms. Especially, we apply Lemma A.3.(2) to the middle terms of each lines, and apply LemmaA.3.(3) to the first terms of each lines. ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 43 Cancellation yields = (cid:88) x m = abcdeb,d,e (cid:54) = ∅ c = ∅ C ( a b (cid:100) ( c d (cid:100) e )) A ( b (cid:99) ce ) B ( d (cid:99) e ) + (cid:88) x m = abcdec,d,e (cid:54) = ∅ C ( a bcd (cid:100) e ) A (( b c (cid:100) d ) (cid:99) e ) B ( c (cid:99) de ) − (cid:88) x m = abcdea,c,d (cid:54) = ∅ C ( a (cid:101) bcd e ) A ( a (cid:98) ( b c (cid:100) d )) B ( c (cid:99) de ) − (cid:88) x m = abcdeb,c,e (cid:54) = ∅ C ( a bcd (cid:100) e ) A (( b (cid:101) c d ) (cid:99) e ) B ( ab (cid:98) c )+ (cid:88) x m = abcdea,b,d (cid:54) = ∅ c = ∅ C (( a (cid:101) b c ) (cid:101) d e ) A ( ac (cid:98) d ) B ( a (cid:98) b ) + (cid:88) x m = abcdea,b,c (cid:54) = ∅ C ( a (cid:101) bcd e ) A ( a (cid:98) ( b (cid:101) c d )) B ( ab (cid:98) c ) − (cid:88) x m = abcdeb,d,e (cid:54) = ∅ c = ∅ C ( a b (cid:100) ( c d (cid:100) e )) B ( b (cid:99) ce ) A ( d (cid:99) e ) − (cid:88) x m = abcdec,d,e (cid:54) = ∅ C ( a bcd (cid:100) e ) B (( b c (cid:100) d ) (cid:99) e ) A ( c (cid:99) de )+ (cid:88) x m = abcdea,c,d (cid:54) = ∅ C ( a (cid:101) bcd e ) B ( a (cid:98) ( b c (cid:100) d )) A ( c (cid:99) de )+ (cid:88) x m = abcdeb,c,e (cid:54) = ∅ C ( a bcd (cid:100) e ) B (( b (cid:101) c d ) (cid:99) e ) A ( ab (cid:98) c ) − (cid:88) x m = abcdea,b,d (cid:54) = ∅ c = ∅ C (( a (cid:101) b c ) (cid:101) d e ) B ( ac (cid:98) d ) A ( a (cid:98) b ) − (cid:88) x m = abcdea,b,c (cid:54) = ∅ C ( a (cid:101) bcd e ) B ( a (cid:98) ( b (cid:101) c d )) A ( ab (cid:98) c ) . By rearranging each terms and by calculating the terms with c = ∅ , we have = (cid:88) x m = abdeb,d,e (cid:54) = ∅ C ( a bd (cid:100) e ) A ( b (cid:99) e ) B ( d (cid:99) e ) − (cid:88) x m = abdeb,d,e (cid:54) = ∅ C ( a bd (cid:100) e ) B ( b (cid:99) e ) A ( d (cid:99) e )+ (cid:88) x m = abdea,b,d (cid:54) = ∅ C ( a (cid:101) bd e ) A ( a (cid:98) d ) B ( a (cid:98) b ) − (cid:88) x m = abdea,b,d (cid:54) = ∅ C ( a (cid:101) bd e ) B ( a (cid:98) d ) A ( a (cid:98) b )+ (cid:88) x m = abcdec,d,e (cid:54) = ∅ C ( a bcd (cid:100) e ) A (( b c (cid:100) d ) (cid:99) e ) B ( c (cid:99) de ) − (cid:88) x m = abcdeb,c,e (cid:54) = ∅ C ( a bcd (cid:100) e ) A (( b (cid:101) c d ) (cid:99) e ) B ( ab (cid:98) c ) − (cid:88) x m = abcdec,d,e (cid:54) = ∅ C ( a bcd (cid:100) e ) B (( b c (cid:100) d ) (cid:99) e ) A ( c (cid:99) de ) + (cid:88) x m = abcdeb,c,e (cid:54) = ∅ C ( a bcd (cid:100) e ) B (( b (cid:101) c d ) (cid:99) e ) A ( ab (cid:98) c )+ (cid:88) x m = abcdea,b,c (cid:54) = ∅ C ( a (cid:101) bcd e ) A ( a (cid:98) ( b (cid:101) c d )) B ( ab (cid:98) c ) − (cid:88) x m = abcdea,c,d (cid:54) = ∅ C ( a (cid:101) bcd e ) A ( a (cid:98) ( b c (cid:100) d )) B ( c (cid:99) de ) − (cid:88) x m = abcdea,b,c (cid:54) = ∅ C ( a (cid:101) bcd e ) B ( a (cid:98) ( b (cid:101) c d )) A ( ab (cid:98) c ) + (cid:88) x m = abcdea,c,d (cid:54) = ∅ C ( a (cid:101) bcd e ) B ( a (cid:98) ( b c (cid:100) d )) A ( c (cid:99) de ) . The cancellations occur on the four pairs: 4th and 19th, 6th and 21th, 7th and 16th, 9thand 18th, and also on the four pairs when c (cid:54) = ∅ : 1st and 15th, 3rd and 13th, 10th and 24th, 12thand 22th. Here, the first term is calculated such that (cid:88) x m = abdeb,d,e (cid:54) = ∅ C ( a bd (cid:100) e ) A ( b (cid:99) e ) B ( d (cid:99) e ) = (cid:88) x m = abceb,c,e (cid:54) = ∅ C ( a bc (cid:100) e ) A ( b (cid:99) e ) B ( c (cid:99) e ) = (cid:88) x m = abcdeb,c,e (cid:54) = ∅ d = ∅ C ( a bcd (cid:100) e ) A (( b c (cid:100) d ) (cid:99) e ) B ( c (cid:99) de ) , The second, third and fourth terms are also calculated respectively as − (cid:88) x m = abdeb,d,e (cid:54) = ∅ C ( a bd (cid:100) e ) B ( b (cid:99) e ) A ( d (cid:99) e ) = − (cid:88) x m = abcdeb,c,e (cid:54) = ∅ d = ∅ C ( a bcd (cid:100) e ) B (( b c (cid:100) d ) (cid:99) e ) A ( c (cid:99) de ) , (cid:88) x m = abdea,b,d (cid:54) = ∅ C ( a (cid:101) bd e ) A ( a (cid:98) d ) B ( a (cid:98) b ) = (cid:88) x m = abcdea,c,d (cid:54) = ∅ b = ∅ C ( a (cid:101) bcd e ) A ( a (cid:98) ( b (cid:101) c d )) B ( ab (cid:98) c ) , − (cid:88) x m = abdea,b,d (cid:54) = ∅ C ( a (cid:101) bd e ) B ( a (cid:98) d ) A ( a (cid:98) b ) = − (cid:88) x m = abcdea,c,d (cid:54) = ∅ b = ∅ C ( a (cid:101) bcd e ) B ( a (cid:98) ( b (cid:101) c d )) A ( ab (cid:98) c ) . These computations yield (arit u ( B ) ◦ arit u ( A ) − arit u ( A ) ◦ arit u ( B ))( C )( x m )= (cid:88) x m = abcdec,e (cid:54) = ∅ C ( a bcd (cid:100) e ) A (( b c (cid:100) d ) (cid:99) e ) B ( c (cid:99) de ) − (cid:88) x m = abcdeb,c,e (cid:54) = ∅ C ( a bcd (cid:100) e ) A (( b (cid:101) c d ) (cid:99) e ) B ( ab (cid:98) c ) − (cid:88) x m = abcdec,e (cid:54) = ∅ C ( a bcd (cid:100) e ) B (( b c (cid:100) d ) (cid:99) e ) A ( c (cid:99) de ) + (cid:88) x m = abcdeb,c,e (cid:54) = ∅ C ( a bcd (cid:100) e ) B (( b (cid:101) c d ) (cid:99) e ) A ( ab (cid:98) c )+ (cid:88) x m = abcdea,c (cid:54) = ∅ C ( a (cid:101) bcd e ) A ( a (cid:98) ( b (cid:101) c d )) B ( ab (cid:98) c ) − (cid:88) x m = abcdea,c,d (cid:54) = ∅ C ( a (cid:101) bcd e ) A ( a (cid:98) ( b c (cid:100) d )) B ( c (cid:99) de ) − (cid:88) x m = abcdea,c (cid:54) = ∅ C ( a (cid:101) bcd e ) B ( a (cid:98) ( b (cid:101) c d )) A ( ab (cid:98) c ) + (cid:88) x m = abcdea,c,d (cid:54) = ∅ C ( a (cid:101) bcd e ) B ( a (cid:98) ( b c (cid:100) d )) A ( c (cid:99) de ) ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 45 = (cid:88) x m = afee (cid:54) = ∅ C ( a f (cid:100) e ) (cid:88) f = bcdc (cid:54) = ∅ A (( b c (cid:100) d ) (cid:99) e ) B ( c (cid:99) de ) − (cid:88) f = bcdb,c (cid:54) = ∅ A (( b (cid:101) c d ) (cid:99) e ) B ( ab (cid:98) c ) − (cid:88) x m = afee (cid:54) = ∅ C ( a f (cid:100) e ) (cid:88) f = bcdc (cid:54) = ∅ B (( b c (cid:100) d ) (cid:99) e ) A ( c (cid:99) de ) − (cid:88) f = bcdb,c (cid:54) = ∅ B (( b (cid:101) c d ) (cid:99) e ) A ( ab (cid:98) c ) + (cid:88) x m = afea (cid:54) = ∅ C ( a (cid:101) f e ) (cid:88) f = bcdc (cid:54) = ∅ A ( a (cid:98) ( b (cid:101) c d )) B ( ab (cid:98) c ) − (cid:88) f = bcdc,d (cid:54) = ∅ A ( a (cid:98) ( b c (cid:100) d )) B ( c (cid:99) de ) − (cid:88) x m = afea (cid:54) = ∅ C ( a (cid:101) f e ) (cid:88) f = bcdc (cid:54) = ∅ B ( a (cid:98) ( b (cid:101) c d )) A ( ab (cid:98) c ) − (cid:88) f = bcdc,d (cid:54) = ∅ B ( a (cid:98) ( b c (cid:100) d )) A ( c (cid:99) de ) = (cid:88) x m = afee (cid:54) = ∅ C ( a f (cid:100) e ) (cid:88) f = bcdc,d (cid:54) = ∅ A (( b c (cid:100) d ) (cid:99) e ) B ( c (cid:99) d ) − (cid:88) f = bcdb,c (cid:54) = ∅ A (( b (cid:101) c d ) (cid:99) e ) B ( b (cid:98) c ) + (cid:88) f = bcc (cid:54) = ∅ A ( b (cid:99) e ) B ( c (cid:99) e ) − (cid:88) x m = afee (cid:54) = ∅ C ( a f (cid:100) e ) (cid:88) f = bcdc,d (cid:54) = ∅ B (( b c (cid:100) d ) (cid:99) e ) A ( c (cid:99) d ) − (cid:88) f = bcdb,c (cid:54) = ∅ B (( b (cid:101) c d ) (cid:99) e ) A ( b (cid:98) c ) + (cid:88) f = bcc (cid:54) = ∅ B ( b (cid:99) e ) A ( c (cid:99) e ) + (cid:88) x m = afea (cid:54) = ∅ C ( a (cid:101) f e ) (cid:88) f = bcdb,c (cid:54) = ∅ A ( a (cid:98) ( b (cid:101) c d )) B ( b (cid:98) c ) − (cid:88) f = bcdc,d (cid:54) = ∅ A ( a (cid:98) ( b c (cid:100) d )) B ( c (cid:99) d ) + (cid:88) f = cdc (cid:54) = ∅ A ( a (cid:98) d ) B ( a (cid:98) c ) − (cid:88) x m = afea (cid:54) = ∅ C ( a (cid:101) f e ) (cid:88) f = bcdb,c (cid:54) = ∅ B ( a (cid:98) ( b (cid:101) c d )) A ( b (cid:98) c ) − (cid:88) f = bcdc,d (cid:54) = ∅ B ( a (cid:98) ( b c (cid:100) d )) A ( c (cid:99) d ) + (cid:88) f = cdc (cid:54) = ∅ B ( a (cid:98) d ) A ( a (cid:98) c ) . By using Lemma A.3.(2), (3) and Lemma A.1, we have = (cid:88) x m = afee (cid:54) = ∅ C ( a f (cid:100) e ) { (arit u ( B )( A ))( f (cid:99) e ) + ( A × B )( f (cid:99) e ) − (arit u ( A )( B ))( f (cid:99) e ) − ( B × A )( f (cid:99) e ) } + (cid:88) x m = afea (cid:54) = ∅ C ( a (cid:101) f e ) {− (arit u ( B )( A ))( a (cid:98) f ) + ( B × A )( a (cid:98) f ) + (arit u ( A )( B ))( a (cid:98) f ) − ( A × B )( a (cid:98) f ) } = (cid:88) x m = afee (cid:54) = ∅ C ( a f (cid:100) e )ari u ( A, B )( f (cid:99) e ) − (cid:88) x m = afea (cid:54) = ∅ C ( a (cid:101) f e )ari u ( A, B )( a (cid:98) f )= (arit u (ari u ( A, B ))( C ))( x m ) . Therefore, we obtain the claim. (cid:3) Definition A.5 ([Ec11] (2.46)) . We consider a binary operation preari u : ARI(Γ) ⊗ → ARI(Γ) which is defined by preari u ( A, B ) := arit u ( B )( A ) + A × B for A, B ∈ ARI(Γ) .Then we have ari u ( A, B ) = preari u ( A, B ) − preari u ( B, A ) . Proposition A.6. The pair (ARI(Γ) , preari u ) forms a pre-Lie algebra, i.e, thefollowing formula holds: preari u ( A, preari u ( B, C )) − preari u (preari u ( A, B ) , C )= preari u ( A, preari u ( C, B )) − preari u (preari u ( A, C ) , B ) for A, B, C ∈ ARI(Γ) .Proof. We have { preari u ( A, preari u ( B, C )) − preari u (preari u ( A, B ) , C ) }− { preari u ( A, preari u ( C, B )) − preari u (preari u ( A, C ) , B ) } = arit u (preari u ( B, C ))( A ) + A × preari u ( B, C ) − arit u ( C )(preari u ( A, B )) − preari u ( A, B ) × C − arit u (preari u ( C, B ))( A ) − A × preari u ( C, B ) + arit u ( B )(preari u ( A, C )) + preari u ( A, C ) × B = arit u (arit u ( C )( B ))( A ) + arit u ( B × C )( A ) + A × (arit u ( C )( B )) + A × ( B × C ) − arit u ( C )(arit u ( B )( A )) − arit u ( C )( A × B ) − arit u ( B )( A ) × C − ( A × B ) × C − arit u (arit u ( B )( C ))( A ) − arit u ( C × B )( A ) − A × (arit u ( B )( C )) − A × ( C × B )+ arit u ( B )(arit u ( C )( A )) + arit u ( B )( A × C ) + arit u ( C )( A ) × B + ( A × C ) × B. By using associativity of (ARI(Γ) , × ) and using Lemma 1.10, we get = arit u (arit u ( C )( B ))( A ) + arit u ( B × C )( A ) − arit u ( C )(arit u ( B )( A )) − arit u (arit u ( B )( C ))( A ) − arit u ( C × B )( A ) + arit u ( B )(arit u ( C )( A ))= arit u (ari u ( B, C ))( A ) − { arit u ( C ) ◦ arit u ( B )( A ) − arit u ( B ) ◦ arit u ( C )( A ) } . Therefore, by using Proposition A.4, we obtain the claim. (cid:3) Proof of Proposition 1.12. It is sufficient to prove the Jacobi identity ari u (ari u ( A, B ) , C ) + ari u (ari u ( B, C ) , A ) + ari u (ari u ( C, A ) , B ) = 0 for A, B, C ∈ ARI(Γ) . By the relationship between ari u and preari u , we calculate ari u (ari u ( A, B ) , C ) + ari u (ari u ( B, C ) , A ) + ari u (ari u ( C, A ) , B )= preari u (ari u ( A, B ) , C ) − preari u ( C, ari u ( A, B ))+ preari u (ari u ( B, C ) , A ) − preari u ( A, ari u ( B, C ))+ preari u (ari u ( C, A ) , B ) − preari u ( B, ari u ( C, A ))= preari u (preari u ( A, B ) − preari u ( B, A ) , C ) − preari u ( C, preari u ( A, B ) − preari u ( B, A ))+ preari u (preari u ( B, C ) − preari u ( C, B ) , A ) − preari u ( A, preari u ( B, C ) − preari u ( C, B ))+ preari u (preari u ( C, A ) − preari u ( A, C ) , B ) − preari u ( B, preari u ( C, A ) − preari u ( A, C )) . By Proposition A.6, it is equal to 0. So we obtain the Jacobi identity. (cid:3) ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 47 A.2. Proof of Proposition 1.13. We give a proof Proposition 1.13 which claimsthat ARI(Γ) al forms a Lie algebra.We show the following key lemma in this section. Lemma A.7. For ω, η, α , . . . , α r ∈ X • Z , we have (A.1) Sh Ç ω ; ηα · · · α r å = (cid:88) ω = ω ··· ω r η = η ··· η r Sh Ç ω ; η α å · · · Sh Ç ω r ; η r α r å . where ω , . . . , ω r , η , . . . , η r run over X • Z .Proof. We consider the deconcatenation coproduct ∆ : A X → A ⊗ X defined by ∆( ω ) := (cid:88) ω = ω ω ω ,ω ∈ X • Z ω ⊗ ω for ω ∈ X • Z . We recursively define Q -linear maps ∆ r : A X → A ⊗ rX by ∆ := ∆ andfor r (cid:62) r := (Id ⊗ · · · ⊗ Id (cid:124) (cid:123)(cid:122) (cid:125) r − ⊗ ∆) ◦ ∆ r − . It is clear that ∆ r is an algebra homomorphism , i.e, for ω, η ∈ X • Z , we have ∆ r ( ω X η ) = ∆ r ( ω ) X ∆ r ( η ) . By expanding the above both sides and taking the coefficient of α ⊗ · · · ⊗ α r for α , . . . , α r ∈ X • Z , we obtain the claim. (cid:3) Proof of Proposition 1.13. Because we have ari( A, B ) = arit( B )( A ) − arit( A )( B ) + [ A, B ] , it is sufficient to prove the following two formulae for A, B ∈ ARI(Γ) al : [ A, B ] ∈ ARI(Γ) al , (A.2) arit( B )( A ) ∈ ARI(Γ) al . (A.3)Firstly, we prove (A.2). Let p, q (cid:62) and put ω = (cid:0) x , ..., x p σ , ..., σ p (cid:1) and η = (cid:0) x p +1 , ..., x p + q σ p +1 , ..., σ p + q (cid:1) .For our simplicity, we denote Sh ( M )( ω ; η ) := (cid:88) α ∈ X • Z Sh Ç ω ; ηα å M p + q ( α ) . Then we have Sh ( A × B )( ω ; η ) = (cid:88) α ∈ X • Z Sh Ç ω ; ηα å (cid:88) α = α α A ( α ) B ( α ) = (cid:88) α ,α ∈ X • Z Sh Ç ω ; ηα α å A ( α ) B ( α ) . Note that the product X of A X induces the product of A ⊗ rX (we also denote this product tothe same symbol X ) as ( ω ⊗ · · · ⊗ ω r ) X ( η ⊗ · · · ⊗ η r ) := ( ω X η ) ⊗ · · · ⊗ ( ω r X η r ) for any ω i , η i ∈ X Z • . By using Lemma A.7 for r = 2 , we get = (cid:88) α ,α ∈ X • Z (cid:88) ω = ω ω η = η η Sh Ç ω ; η α å Sh Ç ω ; η α å A ( α ) B ( α )= (cid:88) ω = ω ω η = η η (cid:88) α ∈ X • Z Sh Ç ω ; η α å A ( α ) (cid:88) α ∈ X • Z Sh Ç ω ; η α å B ( α ) . By using alternality of A , we calculate = (cid:88) ω = ω ω ω (cid:54) = ∅ A ( ω ) (cid:88) α ∈ X • Z Sh Ç ω ; ηα å B ( α ) + (cid:88) η = η η η (cid:54) = ∅ A ( η ) (cid:88) α ∈ X • Z Sh Ç ω ; η α å B ( α ) . By using alternality of B , we obtain = A ( ω ) B ( η ) + A ( η ) B ( ω ) . Because we have [ A, B ] = A × B − B × A , we get Sh ([ A, B ])( ω ; η ) = 0 , that is, weobtain (A.2).Secondly, we prove (A.3). We remark that, for α ∈ X • Z with l ( α ) (cid:62) , we have arit( A )( B )( α )= (cid:88) α = α α α α ,α (cid:54) = ∅ A ( α α (cid:100) α ) B ( α (cid:99) α ) − (cid:88) α = α α α α ,α (cid:54) = ∅ A ( α (cid:101) α α ) B ( α (cid:98) α )= (cid:88) α = α α α α (cid:54) = ∅ A ( α α (cid:100) α ) B ( α (cid:99) α ) − (cid:88) α = α α α α (cid:54) = ∅ A ( α (cid:101) α α ) B ( α (cid:98) α )= (cid:88) α = α α xα α i ∈ X • Z ,x ∈ X Z A ( α α (cid:100) xα ) B ( α (cid:99) x ) − (cid:88) α = α xα α α i ∈ X • Z ,x ∈ X Z A ( α x (cid:101) α α ) B ( x (cid:98) α ) . Then we calculate Sh (arit( B )( A ))( ω ; η )= (cid:88) α ∈ X • Z Sh Ç ω ; ηα å (cid:88) α = α α xα α i ∈ X • Z ,x ∈ X Z A ( α α (cid:100) xα ) B ( α (cid:99) x ) − (cid:88) α = α xα α α i ∈ X • Z ,x ∈ X Z A ( α x (cid:101) α α ) B ( x (cid:98) α ) = (cid:88) α ,α ,α ∈ X • Z x ∈ X Z ® Sh Ç ω ; ηα α xα å A ( α α (cid:100) xα ) B ( α (cid:99) x ) − Sh Ç ω ; ηα xα α å A ( α x (cid:101) α α ) B ( x (cid:98) α ) ´ . ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 49 By using Lemma A.7, we have = (cid:88) α ,α ,α ∈ X • Z x ∈ X Z (cid:88) ω = ω ω ω (cid:48) ω η = η η η (cid:48) η Sh Ç ω ; η α å Sh Ç ω ; η α å Sh Ç ω (cid:48) ; η (cid:48) x å Sh Ç ω ; η α å A ( α α (cid:100) xα ) B ( α (cid:99) x ) − (cid:88) ω = ω ω (cid:48) ω ω η = η η (cid:48) η η Sh Ç ω ; η α å Sh Ç ω (cid:48) ; η (cid:48) x å Sh Ç ω ; η α å Sh Ç ω ; η α å A ( α x (cid:101) α α ) B ( x (cid:98) α ) . Here, if Sh (cid:0) ω ; η α (cid:1) (cid:54) = 0 holds for α ∈ X • Z , then all letters appearing in α matchwith all ones appearing in ω and η . So we have α (cid:100) x = ( ω ,η ) (cid:100) x and x (cid:101) α = x (cid:101) ( ω ,η ) for α ∈ X • Z and we continue = (cid:88) α ,α ∈ X • Z x ∈ X Z (cid:88) ω = ω ω ω (cid:48) ω η = η η η (cid:48) η Sh Ç ω ; η α å Sh Ç ω (cid:48) ; η (cid:48) x å Sh Ç ω ; η α å A ( α ( ω ,η ) (cid:100) xα ) Sh ( B )( ω (cid:99) x ; η (cid:99) x ) − (cid:88) ω = ω ω (cid:48) ω ω η = η η (cid:48) η η Sh Ç ω ; η α å Sh Ç ω (cid:48) ; η (cid:48) x å Sh Ç ω ; η α å A ( α x (cid:101) ( ω ,η ) α ) Sh ( B )( x (cid:98) ω ; x (cid:98) η ) . Because B ∈ ARI(Γ) al , we have Sh ( B )( ∅ ; ∅ ) = 0 and Sh ( B )( ω (cid:99) x ; η (cid:99) x ) = Sh ( B )( x (cid:98) ω ; x (cid:98) η ) =0 for ω , η (cid:54) = ∅ . So we have = (cid:88) α ,α ∈ X • Z x ∈ X Z (cid:88) ω = ω ω ω (cid:48) ω η = η η (cid:48) η ω (cid:54) = ∅ Sh Ç ω ; η α å Sh Ç ω (cid:48) ; η (cid:48) x å Sh Ç ω ; η α å A ( α ω (cid:100) xα ) B ( ω (cid:99) x ) − (cid:88) ω = ω ω (cid:48) ω ω η = η η (cid:48) η ω (cid:54) = ∅ Sh Ç ω ; η α å Sh Ç ω (cid:48) ; η (cid:48) x å Sh Ç ω ; η α å A ( α x (cid:101) ω α ) B ( x (cid:98) ω )+ (cid:88) ω = ω ω (cid:48) ω η = η η η (cid:48) η η (cid:54) = ∅ Sh Ç ω ; η α å Sh Ç ω (cid:48) ; η (cid:48) x å Sh Ç ω ; η α å A ( α η (cid:100) xα ) B ( η (cid:99) x ) − (cid:88) ω = ω ω (cid:48) ω η = η η (cid:48) η η η (cid:54) = ∅ Sh Ç ω ; η α å Sh Ç ω (cid:48) ; η (cid:48) x å Sh Ç ω ; η α å A ( α x (cid:101) η α ) B ( x (cid:98) η ) . Here, Sh (cid:0) ω (cid:48) ; η (cid:48) x (cid:1) (cid:54) = 0 holds for x ∈ X Z if and only if ( ω (cid:48) , η (cid:48) ) = ( x, ∅ ) or ( ∅ , x ) . So weget = (cid:88) α ,α ∈ X • Z x ∈ X Z (cid:88) ω = ω ω xω η = η η ω (cid:54) = ∅ Sh Ç ω ; η α å Sh Ç ω ; η α å A ( α ω (cid:100) xα ) B ( ω (cid:99) x ) − (cid:88) ω = ω xω ω η = η η ω (cid:54) = ∅ Sh Ç ω ; η α å Sh Ç ω ; η α å A ( α x (cid:101) ω α ) B ( x (cid:98) ω )+ (cid:88) ω = ω xω η = η η η η (cid:54) = ∅ Sh Ç ω ; η α å Sh Ç ω ; η α å { A ( α η (cid:100) xα ) B ( η (cid:99) x ) − A ( α x (cid:101) η α ) B ( x (cid:98) η ) } + (cid:88) α ,α ∈ X • Z x ∈ X Z (cid:88) ω = ω ω ω η = η xη ω (cid:54) = ∅ Sh Ç ω ; η α å Sh Ç ω ; η α å { A ( α ω (cid:100) xα ) B ( ω (cid:99) x ) − A ( α x (cid:101) ω α ) B ( x (cid:98) ω ) } + (cid:88) ω = ω ω η = η η xη η (cid:54) = ∅ Sh Ç ω ; η α å Sh Ç ω ; η α å A ( α η (cid:100) xα ) B ( η (cid:99) x ) − (cid:88) ω = ω ω η = η xη η η (cid:54) = ∅ Sh Ç ω ; η α å Sh Ç ω ; η α å A ( α x (cid:101) η α ) B ( x (cid:98) η ) . ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 51 Because x runs over X Z , we get ω (cid:100) x = x (cid:101) ω and η (cid:100) x = x (cid:101) η and η (cid:99) x = x (cid:98) η and ω (cid:99) x = x (cid:98) ω . Hence we calculate = (cid:88) α ,α ∈ X • Z x ∈ X Z (cid:88) ω = ω ω xω η = η η ω (cid:54) = ∅ Sh Ç ω ; η α å Sh Ç ω ; η α å A ( α ω (cid:100) xα ) B ( ω (cid:99) x ) − (cid:88) ω = ω xω ω η = η η ω (cid:54) = ∅ Sh Ç ω ; η α å Sh Ç ω ; η α å A ( α ω (cid:100) xα ) B ( ω (cid:99) x ) + (cid:88) α ,α ∈ X • Z x ∈ X Z (cid:88) ω = ω ω η = η η xη η (cid:54) = ∅ Sh Ç ω ; η α å Sh Ç ω ; η α å A ( α η (cid:100) xα ) B ( η (cid:99) x ) − (cid:88) ω = ω ω η = η xη η η (cid:54) = ∅ Sh Ç ω ; η α å Sh Ç ω ; η α å A ( α η (cid:100) xα ) B ( η (cid:99) x ) . For ω , ω , ω , x with ω = ω ω xω , by using Lemma A.7 with r = 3 and ω = ω ω (cid:100) xω and α = ω (cid:100) x , we have Sh Ç ω ω (cid:100) xω ; ηα ω (cid:100) xα å = (cid:88) ω ω (cid:100) xω = ω (cid:48) ω (cid:48) ω (cid:48) η = η η η Sh Ç ω (cid:48) ; η α å Sh Ç ω (cid:48) ; η ω (cid:100) x å Sh Ç ω (cid:48) ; η α å . Because η = (cid:0) x p +1 , ..., x p + q σ p +1 , ..., σ p + q (cid:1) and ω (cid:54) = ∅ , the letter ω (cid:100) x ∈ X Z does not appear in η .So for any word η such that η = η η η , we get η (cid:54) = ω (cid:100) x . Hence, Sh (cid:0) ω (cid:48) ; η ω (cid:100) x (cid:1) (cid:54) = 0 holds if and only if ω (cid:48) = ω (cid:100) x and η = ∅ . So we have = (cid:88) η = η η Sh Ç ω ; η α å Sh Ç ω ; η α å . Similarly, we get (cid:88) ω = ω ω Sh Ç ω ; η α å Sh Ç ω ; η α å = Sh Ç ω ; η η (cid:100) xη α ω (cid:100) xα å . So by using these, we calculate Sh (arit( B )( A ))( ω ; η )= (cid:88) x ∈ X Z (cid:88) ω = ω ω xω ω (cid:54) = ∅ Sh ( A )( ω ω (cid:100) xω ; η ) B ( ω (cid:99) x ) − (cid:88) ω = ω xω ω ω (cid:54) = ∅ Sh ( A )( ω ω (cid:100) xω ; η ) B ( ω (cid:99) x ) + (cid:88) x ∈ X Z (cid:88) η = η η xη η (cid:54) = ∅ Sh ( A )( ω ; η η (cid:100) xη ) B ( η (cid:99) x ) − (cid:88) η = η xη η η (cid:54) = ∅ Sh ( A )( ω ; η η (cid:100) xη ) B ( η (cid:99) x ) . Lastly, similarly to footnote 3 in Definition 1.9, all letters appearing in twowords ω ω (cid:100) xω and η (resp. ω and η η (cid:100) xη ) are algebraically independent over Q ,so by Remark 1.5, the component Sh ( A )( ω ω (cid:100) xω ; η ) (resp. Sh ( A )( ω ; η η (cid:100) xη ) ) iswell-defined. Hence, by using alternality of A , we obtain (A.3). (cid:3) A.3. Proof of Proposition 1.21. We give a proof Proposition 1.21 which claimsthat ARI(Γ) al / al forms a filtered Lie subalgebra of ARI(Γ) al under the ari u -bracket.For A, B ∈ ARI(Γ) al / al , it is enough to prove ari u ( A, B ) ∈ ARI(Γ) al / al , that is, ari u ( A, B ) ∈ ARI(Γ) al , (A.4) swap(ari u ( A, B )) ∈ ARI(Γ) al , (A.5) ari u ( A, B ) (cid:0) x σ (cid:1) = ari u ( A, B ) (cid:16) − x σ − (cid:17) . (A.6)Because ARI(Γ) al forms a Lie algebra under the ari u -bracket, (A.4) is obvious. Bythe definition of ari u , (A.6) is also clear. By Lemma 1.27 and Proposition 3.11, wehave swap(ari u ( A, B )) = ari v (swap( A ) , swap( B )) . Since we have swap( A ) , swap( B ) ∈ ARI(Γ) al for A , B ∈ ARI(Γ) al / al , we get swap(ari u ( A, B )) ∈ ARI(Γ) al by Proposition 1.19. Thus we obtain (A.5). (cid:3) Appendix B. Multiple polylogarithms at roots of unity In this appendix we recall how ARI(Γ) al / al and D (Γ) •• are related to multiplepolylogarithms at roots of unity.Multiple polylogarithm is the several variable complex function which is definedby the following power series Li n ,...,n r ( z , . . . , z r ) := (cid:88) Zag (cid:0) u , ..., u m (cid:15) , ..., (cid:15) m (cid:1) = (cid:88) n ,...,n m > Li X n ,...,n m ( (cid:15) (cid:15) , . . . , (cid:15) m − (cid:15) m , (cid:15) m ) u n − ( u + u ) n − · · · ( u + · · · + u m ) n m − Zig (cid:0) (cid:15) , ..., (cid:15) m v , ..., v m (cid:1) = (cid:88) n ,...,n m > Li ∗ n m ,...,n ( (cid:15) m , . . . , (cid:15) ) v n − · · · v n m − m where Li X n ,...,n m ( (cid:15) , . . . , (cid:15) m ) and Li ∗ n ,...,n m ( (cid:15) , . . . , (cid:15) m ) mean the shuffle regular-ization and the harmonic (stuffle) regularization of Li n ,...,n m ( (cid:15) , . . . , (cid:15) m ) respec-tively (cf. [AK]). Particularly Li X n ,...,n m ( (cid:15) , . . . , (cid:15) m ) = Li ∗ n ,...,n m ( (cid:15) , . . . , (cid:15) m ) =Li n ,...,n m ( (cid:15) , . . . , (cid:15) m ) when ( n m , (cid:15) m ) (cid:54) = (1 , . In [Ec03, (37)] and [Ec11, (1.27)], itis explained that they are related as follows(B.1) Zig = Mini × swap(Zag) (see (1.7) for swap ). Here Mini = { Mini (cid:0) u , ..., u m (cid:15) , ..., (cid:15) m (cid:1) } m is the mould defined by Mini (cid:0) (cid:15) , ..., (cid:15) m v , ..., v m (cid:1) = ® Mono m when ( (cid:15) , . . . , (cid:15) m ) = (1 , . . . , , otherwisewith (cid:80) ∞ r =2 Mono r · t r := exp { (cid:80) ∞ k =2 ( − k − ζ ( k ) k t k } (cf. [Ec11, (1.30)]).Goncharov’s arguments in [G01b] suggest us to express them as Zag (cid:0) u , ..., u m (cid:15) , ..., (cid:15) m (cid:1) = reg X (cid:90) GARI(Γ) as / is . The authors are not aware of its defini-tion but expect that it means the symmetrality for Zag and the symmetrilty for Zig ([Ec11, §§1.1–1.2]), which looks corresponding to the shuffle and the harmonicproduct among multiple L -values. In [Ec11, §4.7] it is directed to combine a re-lated group GARI(Γ) as / is and its Lie algebra ARI(Γ) al / il with a bigraded variant GARI(Γ) as / as and ARI(Γ) al / al (cf. Definition 1.20) under maps adgari(pal) and adari(pal) (cf. [Ec11, (2.54), (2.55), and §4.2]):(B.2) GARI(Γ) as / asadgari(pal) (cid:47) (cid:47) logari (cid:15) (cid:15) GARI(Γ) as / islogari (cid:15) (cid:15) ARI(Γ) al / al adari(pal) (cid:47) (cid:47) ARI(Γ) al / il . While the cyclotomic analogue of Drinfeld’s KZ-associator ([D]) which is con-structed from the KZ-like differential equation in ˆ A C over P ( C ) \ { , µ N , ∞} witha complex variable s dds H ( s ) = Ñ xs + (cid:88) ξ ∈ µ N y ξ ξs − é H ( s ) is introduced and discussed in [R02, En]. It is a non-commutative formal powerseries Φ N KZ ∈ ˆ A C with Γ = µ N whose coefficients are given by multiple L -values. In particular, the coefficient of x n r − y (cid:15) r x n r − − y (cid:15) r − (cid:15) r · · · x n − y (cid:15) ··· (cid:15) r is ( − r Li X n ,...,n r ( (cid:15) , . . . , (cid:15) r ) . Definition 2.4 enables us to calculate a relation be-tween the associator Φ N KZ and the mould Zag as follows ma (Φ N KZ ) − = (cid:110) Zag (cid:16) − u ,..., − u m (cid:15) − , ... , (cid:15) − m (cid:17)(cid:111) m ∈ M ( F ; µ N ) . (B.3)Put (cid:101) Φ N KZ = Φ N KZ ( x, ( − y σ )) and (cid:101) Φ KZ , corr = exp ® (cid:80) n (cid:62) − n − n Li n (1) y n ´ ∈ ˆ A C anddefine(B.4) (cid:101) Φ N KZ , ∗ := (cid:101) Φ KZ , corr · q ( π Y ( (cid:101) Φ N KZ )) (for π Y and q , see §2.1). It is shown in [R02] that (cid:101) Φ N KZ is group-like with respect tothe shuffle (deconcatenation) coproduct of ˆ A C and (cid:101) Φ N KZ , ∗ is so with respect to theharmonic coproduct of Im π Y . They correspond to the shuffle and the harmonicproduct among multiple L -values respectively. The regularized double shuffle rela-tions (the shuffle and the harmonic products and the regularization relations (B.4))are the defining equations of his torsor DMR µ for µ ∈ C × which contains (cid:101) Φ N KZ as a specific point when µ = 2 π √− . It is equipped with a free and transitiveaction of the group DMR . Its associated Lie algebra, which he denotes by dmr ,is a filtered graded Lie algebra defined by the regularized double shuffle relationsmodulo products. Our dihedral Lie algebra D ( µ N ) •• should be called its bigradedvariant, defined by ‘their highest depth parts’ of the relations. By definition, itcontains the associated graded gr dmr . By translating Ecalle’s pictures includingthe diagram (B.2) to this setting, we might learn more enriched structures on theseLie algebras. References [AET] Alekseev, A., Enriquez, B. and Torossian, C., Drinfeld associators, braid groups and ex-plicit solutions of the Kashiwara-Vergne equations , Publ. Math. Inst. Hautes Études Sci. No. (2010), 143–189.[AKKN] Alekseev, A., Kawazumi. N, Kuno. Y. and Naef, F. The Goldman-Turaev Lie bialgebrain genus zero and the Kashiwara-Vergne problem , Advances in Mathematics, , 1-53 (2018).[AT] Alekseev, A. and Torossian, C., The Kashiwara-Vergne conjecture and Drinfeld’s associa-tors , Ann. of Math. (2) (2012), no. 2, 415–463.[AK] Arakawa, T., Kaneko, M., On multiple L -values , J. Math. Soc. Japan 56 (2004), no.4,967–991.[C] Cresson. J., Calcul moulien , Ann. Fac. Sci. Toulouse Math. (6) (2009), no. 2, 307–395.[D] V. Drinfeld, On quasitriangular quasi-Hopf algebras and on a group that is closely connectedwith Gal( Q / Q ) , Leningrad Math. J. 2 (1991), no. 4, 829–860.[Ec81] Ecalle, J., Les fonctions résurgentes. Tome I et II , Publications Mathématiques d’Orsay , 6. Université de Paris-Sud, Département de Mathématique, Orsay, 1981.[Ec03] Ecalle, J., ARI/GARI, la dimorphie et l’arithmétique des multizêtas: un premier bilan , J.Théeor. Nombres Bordeaux (2003), no. 2, 411–478.[Ec11] Ecalle, J., The flexion structure and dimorphy: flexion units, singulators, generators, andthe enumeration of multizeta irreducibles , With computational assistance from S. Carr. CRMSeries, , Asymptotics in dynamics, geometry and PDEs; generalized Borel summation. Vol. II,27–211, Ed. Norm., Pisa, 2011.[En] Enriquez, B., Quasi-reflection algebras and cyclotomic associators , Selecta Math.(N.S.) 13(2007), no. 3, 391–463. The word DMR stands for the French ‘double mélange et régularisation’. ASHIWARA-VERGNE AND DIHEDRAL LIE ALGEBRAS 55 [F] Furusho, H., Around associators , Automorphic forms and Galois representations. Vol. 2, 105–117, London Math. Soc. Lecture Note Ser., , Cambridge Univ. Press, Cambridge, 2014.[G01a] Goncharov, A. B., The dihedral Lie algebras and Galois symmetries of π ( l )1 ( P − ( { , ∞}∪ µ N )) , Duke Math. J. (2001), no. 3, 397–487.[G01b] Goncharov, A. B., Multiple polylogarithms and mixed Tate motives , preprint, arXiv:math/0103059 .[M] Maassarani, M., Bigraded Lie algebras related to MZVs , preprint, arXiv:1907.07200 .[R00] Racinet, G., Séries génératrices non-commutatives de polyzêtas et associateurs de Drinfeld ,Ph.D. desseratation, Paris, France, 2000.[R02] Racinet, G., Doubles mélanges des polylogarithmes multiples aux racines de l’unité , Publ.Math. Inst. Hautes Études Sci. (2002), 185–231.[RS] Raphael, E., Schneps, L., On linearised and elliptic versions of the Kashiwara-Vergne Liealgebra , arXiv:1706.08299v1 , preprint.[SaSch] Salerno, A., Schnepps, L., Mould theory and the double shuffle Lie algebra structure ,Periods in Quantum Field Theory and Arithmetic, Springer Proceedings in Mathematics &Statistics, vol (2020), Springer, 399–430.[Sau] Sauzin, D., Mould expansions for the saddle-node and resurgence monomials , Renormaliza-tion and Galois theories, 83–163, IRMA Lect. Math. Theor. Phys., , Eur. Math. Soc., Zürich,2009.[Sch12] Schneps, L., Double shuffle and Kashiwara-Vergne Lie algebras , J. Algebra (2012),54–74.[Sch15] Schneps, L., ARI, GARI, ZIG and ZAG: An introduction to Ecalle’s theory of multiplezeta values , arXiv:1507.01534 , preprint. Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya,464-8602, Japan E-mail address : [email protected] E-mail address ::