ACPMS On Kashiwara-Vergne bigraded Lie algebra in mould theory Nao - PowerPoint PPT Presentation

ACPMS On Kashiwara-Vergne bigraded Lie algebra in mould theory Nao Komiyama Nagoya University 27/11/2020 Nao KOMIYAMA ACPMS 27/11/2020 1 / 35

Contents Introduction 1 Kashiwara-Vergne group Kashiwara-Vergne Lie algebra Mould 2 Definition of mould Lie algebra ARI (Γ) of moulds Lie algebra 3 Lie subalgebras of ARI (Γ) Kashiwara-Vergne bigraded Lie algebra lkrv (Γ) •• Nao KOMIYAMA ACPMS 27/11/2020 2 / 35

Section 1 Introduction Nao KOMIYAMA ACPMS 27/11/2020 3 / 35

Introduction ∼ Kashiwara-Vergne group ∼ Kashiwara and Vergne proposed a conjecture related to the Campbell-Baker-Hausdorff series in the following paper: The Campbell-Hausdorff formula and invariant hyperfunctions , Invent. Math. 47 (1978), no. 3, 249-272. There is one of the formulations of the above KV conjecture by Alekseev, Enriquez and Torossian. Notation • Let F 2 be the completed free Lie algebra with two variables X 0 and X 1 . • Denote by U F 2 = k ⟨⟨ X 0 , X 1 ⟩⟩ the non-commutative formal power series ring defined as the universal enveloping algebra of F 2 . • Put exp F 2 to be the image of F 2 under the map exp : F 2 → U F 2 defined by x n ∑ exp( x ) := ( x ∈ F 2 ) . n ! n ≥ 0 Nao KOMIYAMA ACPMS 27/11/2020 4 / 35

Introduction ∼ Kashiwara-Vergne group ∼ Generalized Kashiwara-Vergne problem ( [Alekseev-Enriquez-Torossian] ) Find a group automorphism P : exp F 2 → exp F 2 such that P ∈ TAut exp F 2 and P satisfies P ( e X 0 e X 1 ) = e X 0 + X 1 and the coboundary Jacobian condition δ ◦ J ( P ) = 0 . Here, P ∈ TAut exp F 2 means that P is in Aut exp F 2 such that P ( e X 0 ) = p 1 e X 0 p − 1 P ( e X 1 ) = p 2 e X 1 p − 1 and 1 2 for some p 1 , p 2 ∈ exp F 2 . J stands for the Jacobian cocycle and δ means the differential map. Nao KOMIYAMA ACPMS 27/11/2020 5 / 35

Introduction ∼ Kashiwara-Vergne group ∼ Definition ([A-E-T; 2010], [A-T; 2012]) The Kashiwara-Vergne group KRV is defined to be the set of P ∈ TAut exp F 2 which satisfies P ( e X 0 e X 1 ) = e X 0 + X 1 and the coboundary Jacobian condition δ ◦ J ( P ) = 0. Remark The set KRV forms a subgroup of Aut exp F 2 . Nao KOMIYAMA ACPMS 27/11/2020 6 / 35

Introduction ∼ Kashiwara-Vergne group ∼ We denote KRV 0 to be a subgroup of KRV consisting of P without linear terms in p 1 and p 2 . There are the following inclusions. Theorem ([A-E-T; 2010], [A-T; 2012]) GRT 1 ⊂ KRV 0 . Theorem ([Schneps; 2012]) DMR 0 ⊂ KRV 0 . Here, GRT 1 is the Grothendieck-Teichm¨ uller group introduced by Drinfel’d, and DMR 0 is the double shuffle group introduced by Racinet. cf.) H. Furusho, Around associators . Automorphic forms and Galois representations, (2014) Vol. 2, 105–117. Nao KOMIYAMA ACPMS 27/11/2020 7 / 35

Introduction ∼ Kashiwara-Vergne Lie algebra ∼ To recall the definition of KV Lie algebra, we prepare some notations. • Let L = ⊕ w ≥ 1 L w be the free graded Lie Q -algebra generated by two variables x and y with deg x = deg y = 1 ( L w is the Q -linear space generated by Lie monomials whose total degree is w ). • The non-commutative polynomial algebra A = Q ⟨ x , y ⟩ is regarded as the universal enveloping algebra of L . • Put Cyc ( A ) to be the Q -linear space generated by cyclic words of A , and put the trace map tr : A ↠ Cyc ( A ) to be the natural projection. Nao KOMIYAMA ACPMS 27/11/2020 8 / 35

Introduction ∼ Kashiwara-Vergne Lie algebra ∼ Definition ([A-E-T; 2010], [A-T; 2012]) The Kashiwara-Vergne (graded) Lie algebra is the graded Q -linear space krv • = ⊕ w ≥ 2 krv w . Here, its degree w -part krv 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 ] + [ y , F ] = 0 (KV1) and α ∈ Q with tr ( G x x + F y y ) = α · tr (( x + y ) w − x w − y w ) (KV2) when we write F = F x x + F y y and G = G x x + G y y in A . We note that such G = G ( F ) uniquely exists for F ∈ L w when w ≥ 2. Nao KOMIYAMA ACPMS 27/11/2020 9 / 35

Introduction ∼ Kashiwara-Vergne Lie algebra ∼ The Lie algebra structure of krv • • Let tder be the set of tangential derivation of L . The derivation D F , G of L defined by x �→ [ x , G ] and y �→ [ y , F ] for some F , G ∈ L . It forms a Lie algebra by the bracket [ D F 1 , G 1 , D F 2 , G 2 ] = D F 1 , G 1 ◦ D F 2 , G 2 − D F 2 , G 2 ◦ D F 1 , G 1 . • We denote sder to be set of special derivations, which are tangential derivations D such that D ( x + y ) = 0. It forms a Lie subalgebra of tder . The Lie algebra structure of krv • is defined to be compatible with that of sder under the embedding F ∈ krv • to D F , G ( F ) ∈ sder . Today’s goal To give a Kashiwara-Vergne bigraded Lie algebra lkrv (Γ) •• . Nao KOMIYAMA ACPMS 27/11/2020 10 / 35

Section 2 Mould Nao KOMIYAMA ACPMS 27/11/2020 11 / 35

Mould ∼ Definition ∼ Early 1980s, the mould was introduced by Ecalle in his paper: Les fonctions r´ esurgentes, Tome I, II, III . In 2000s, he applied the moulds to study of multiple zeta values in papers: • ARI/GARI, la dimorphie et l’arithm´ etique des multizetas , • The flexion structure and dimorphy: flexion units, singulators, generators, and the enumeration of multizeta irreducibles . cf.) Schneps, ARI, GARI, Zig and Zag: An introduction to Ecalle’s theory of multiple zeta values , arXiv:1507.01534. Nao KOMIYAMA ACPMS 27/11/2020 12 / 35

Mould ∼ Definition ∼ ∪ Let Γ be a finite abelian group. We set F := Q ( x 1 , . . . , x m ). m ⩾ 1 Definition ([Furusho-K, Definition 1.1]) A mould on Z ⩾ 0 with values in F is a collection (a sequence) M = ( M m ( x 1 , . . . , x m )) m ∈ Z ⩾ 0 = M 0 ( ∅ ) , M 1 ( x 1 ) , M 2 ( x 1 , x 2 ) , . . . ( ) , with M 0 ( ∅ ) ∈ Q and M m ( x 1 , . . . , x m ) ∈ Q ( x 1 , . . . , x m ) ⊕ Γ ⊕ m for m ⩾ 1, which is described by a summation M m ( x 1 , . . . , x m ) = ( σ 1 ,...,σ m ) ∈ Γ ⊕ m M m ⊕ σ 1 ,...,σ m ( x 1 , . . . , x m ) where each M m σ 1 ,...,σ m ( x 1 , . . . , x m ) ∈ Q ( x 1 , . . . , x m ). Nao KOMIYAMA ACPMS 27/11/2020 13 / 35

Mould ∼ Definition ∼ • We denote the set of all moulds with values in F by M ( F ; Γ). The set M ( F ; Γ) forms a Q -linear space by A + B := ( A m ( x 1 , . . . , x m ) + B m ( x 1 , . . . , x m )) m ∈ Z ⩾ 0 , cA := ( cA m ( x 1 , . . . , x m )) m ∈ Z ⩾ 0 , for A , B ∈ M ( F ; Γ) and c ∈ Q . • We define a product on M ( F ; Γ) by m ∑ ( A × B ) m A i σ 1 ,...,σ i ( x 1 , . . . , x i ) B m − i σ 1 ,...,σ m ( x 1 , . . . , x m ) := σ i +1 ,...,σ m ( x i +1 , . . . , x m ) , i =0 for A , B ∈ M ( F ; Γ) and for m ⩾ 0 and for ( σ 1 , . . . , 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 , 0 , 0 , . . . ). Nao KOMIYAMA ACPMS 27/11/2020 14 / 35

Mould ∼ Example ∼ We give some examples. Example (1). Define A ∈ M ( F ; Γ) by { 0 ( m = 0 , 1) , A m σ 1 ,...,σ m ( x 1 , . . . , x m ) := 1 1 ( x 2 − x 1 ) · · · ( m ≥ 2) , ( x r − x r − 1 ) for m ≥ 0 and for σ 1 , . . . , σ m ∈ Γ. (2). Define B ∈ M ( F ; Γ) by { 0 ( m = 0 , 1) , B m σ 1 ,...,σ m ( x 1 , . . . , x m ) := 1 1 1 x 1 + · · · + x m − ( m ≥ 2) , x 1 + ··· + x m for m ≥ 0 and for σ 1 , . . . , σ m ∈ Γ. Nao KOMIYAMA ACPMS 27/11/2020 15 / 35

Mould ∼ Lie algebra ARI (Γ) ∼ • Put ARI (Γ) := { M ∈ M ( F ; Γ) | M 0 ( ∅ ) = 0 } (this set forms a non-unital subalgebra of ( M ( F ; Γ) , × )). For M ∈ ARI (Γ), we sometimes denote M m σ 1 ,...,σ m ( x 1 , . . . , x m ) by M m ( x 1 , ..., x m ) . σ 1 , ..., σ m • In order to give a Lie algebraic structure to ARI (Γ), we prepare the algebraic formulation: {( x i )} Put X := i ∈ N ,σ ∈ Γ . Let X Z be the set such that σ X Z := { ( u σ ) | u = a 1 x 1 + · · · + a k x k , k ∈ N , a j ∈ Z , σ ∈ Γ } . Let X • Z be the non-commutative free monoid generated by all elements of X Z with the empty word ∅ as the unit. Occasionally we denote each element ω = u 1 · · · u m ∈ X • Z with u 1 , . . . , u m ∈ X Z by ω = ( u 1 , . . . , u m ) as a sequence. The length of ω = u 1 · · · u m is defined to be l ( ω ) := m . Nao KOMIYAMA ACPMS 27/11/2020 16 / 35

Mould ∼ Lie algebra ARI (Γ) ∼ Definition ([F-K, Definition 1.8]) The flexions are the four binary operators ∗ ⌊∗ , ∗⌋ ∗ : X • Z × X • Z → X • ∗ ⌈∗ , ∗⌉ ∗ , Z which are defined by ( ) b 1 + · · · + b n + a 1 , a 2 , . . . , a m β ⌈ α := , σ 1 , σ 2 , . . . , σ m ( ) a 1 , . . . , a m − 1 , a m + b 1 + · · · + b n α ⌉ β := , σ 1 , . . . , σ m − 1 , σ m ( a 1 , . . . , ) a m β ⌊ α := , τ − 1 τ − 1 σ 1 , . . . , σ m n n ( ) a 1 , . . . , a m α ⌋ β := , σ 1 τ − 1 σ m τ − 1 , . . . , 1 1 ∅ ⌈ γ := γ ⌉ ∅ := ∅ ⌊ γ := γ ⌋ ∅ := γ, γ ⌈∅ := ∅⌉ γ := γ ⌊∅ := ∅⌋ γ := ∅ , ( ) ( a 1 ,..., a m b 1 ,..., b n ∈ X • Z ( m , n ⩾ 1) and γ ∈ X • ) for α = , β = Z . σ 1 ,...,σ m τ 1 ,...,τ n Nao KOMIYAMA ACPMS 27/11/2020 17 / 35