A propos dun groupe dassociateurs ui 0 , G.H.E. Duchamp 1 , 4 , V.C. - PowerPoint PPT Presentation

A propos d’un groupe d’associateurs ui 0 , G.H.E. Duchamp 1 , 4 , V.C. B` V. Hoang Ngoc Minh 2 , 4 , K.A. Penson 3 , Q.H. Ngˆ o 5 0 Hue University of Sciences, 77 - Nguyen Hue street - Hue city, Vietnam. 1 Universit´ e Paris 13, 99 avenue Jean-Baptiste Cl´ ement, 93430 Villetaneuse, France. 2 Universit´ e Lille 2, 1, Place D´ eliot, 59024 Lille, France. 3 Universite Paris VI, 75252 Paris Cedex 05, France 4 LIPN-UMR 7030, 99 avenue Jean-Baptiste Cl´ ement, 93430 Villetaneuse, France. 5 University of Hai Phong, 171, Phan Dang Luu, Kien An, Hai Phong, Viet Nam Journ´ ees Nationales de Calcul Formel 22-26 Janvier, 2018, Luminy, France

Outline 1. Introduction 1.1 Zeta functions with several complex indices 1.2 Noncommutative, co-commutative bialgebras 1.3 First structure of polylogarithms and harmonic sums 2. Singular and asymptotic expansions 2.1 Noncommutative generating series and first Abel like theorem for noncommutative generating series 2.2 Actions of the Galois differential group over singular and asymptotic expansions 2.3 Bi-integro differential algebra and second Abel like theorem for noncommutative generating series 3. Polylogarithms, harmonic sums indexed by noncommutative rational series 3.1 Polylogarithms, harmonic sums and rational series 3.2 Constants { γ − s 1 ,..., − s r } ( s 1 ,..., s r ) ∈ N r , r ∈ N 3.3 Candidates for associators with rational coefficients

INTRODUCTION

Zeta functions with several complex indices H r := { ( s 1 , . . . , s r ) ∈ C r |∀ m = 1 , . . . , r , ℜ ( s 1 ) + . . . + ℜ ( s m ) > m } , r ∈ N + . � n − s 1 . . . n − s r ζ ( s 1 , . . . , s r ) = converges for ( s 1 , . . . , s r ) ∈ H r . 1 r n 1 >...> n r > 0 For n ∈ N , z ∈ C , | z | < 1 , ( s 1 , . . . , s r ) ∈ C r , let us define the following functions z n 1 Li s 1 ,..., s r ( z ) � � H s 1 ,..., s r ( n ) z n . Li s 1 ,..., s r ( z ) = and = n s 1 1 . . . n s r 1 − z r n 1 >...> n r > 0 n ≥ 0 Hence, from a theorem by Abel, one has ∀ ( s 1 , . . . , s r ) ∈ H r , ζ ( s 1 , . . . , s r ) = n → + ∞ H s 1 ,..., s r ( n ) = lim lim z → 1 Li s 1 ,..., s r ( z ) . Z := span Q { ζ ( s 1 , . . . , s r ) } ( s 1 ,..., s r ) ∈H r ∩ N r , r ∈ N . These values do appear in the regularization of solutions of the following differential equation with noncommutative indeterminates in X = { x 0 , x 1 } dG = MG , with M = ω 0 x 0 + ω 1 x 1 , ω 0 ( z ) = dz dz ( DE ) z , ω 1 ( z ) = 1 − z . Drinfel’d stated that ( DE ) has a unique solution G 0 (resp. G 1 ), being group-like series, s.t. G 0 ( z ) ∼ 0 e x 0 log( z ) (resp. G 1 ( z ) ∼ 1 e − x 1 log(1 − z ) ). There is then a unique series Φ KZ ∈ R � � X � � , ∆ ⊔ ⊔ (Φ KZ ) = Φ KZ ⊗ Φ KZ , such that G 0 = G 1 Φ KZ . This series is called Drinfel’d associator.

Indexing by words Introducing Y = { y k } k ≥ 1 , Y 0 = Y ∪ { y 0 } and using the correspondences π X x s 1 − 1 x 1 . . . x s r − 1 ( s 1 , . . . , s r ) ∈ N r y s 1 . . . y s r ∈ Y ∗ x 1 ∈ X ∗ x 1 , ↔ ⇋ + 0 0 π Y ( s 1 , . . . , s r ) ∈ N r y s 1 . . . y s r ∈ Y ∗ ↔ 0 , we denote H y s 1 ... y sr := H s 1 ,..., s r and Li x s 1 − 1 x 1 := Li s 1 ,..., s r , x 1 ... x sr − 1 0 0 H − Li − y s 1 ... y sr := H − s 1 ,..., − s r and y s 1 ... y sr := Li − s 1 ,..., − s r , =: ζ ( x s 1 − 1 x 1 . . . x s r − 1 and also ζ ( y s 1 . . . y s r ) := ζ ( s 1 , . . . , s r ) x 1 ) , 0 0 ζ − ( y s 1 . . . y s r ) := ζ ( − s 1 , . . . , − s r ) . The polylogarithms can be viewed as iterated integrals, w.r.t. ω 0 , ω 1 and associated to words in X ∗ : Li s 1 ,..., s r ( z ) = α z 0 ( x s 1 − 1 x 1 . . . x s r − 1 x 1 ), where 0 0 � z � z k − 1 α z α z z 0 (1 X ∗ ) = 1 Ω and z 0 ( x i 1 . . . x i k ) = ω i 1 ( z 1 ) . . . ω i k ( z k ) , z 0 z 0 where ( z 0 , z 1 . . . , z k , z ) is a subdivision of the path z 0 � z in the simply � C − { 0 , 1 } and 1 Ω : Ω → C , mapping z to 1. connected domain Ω := θ 0 := z ∂ z , θ 1 := (1 − z ) ∂ z and ι 0 , ι 1 such that θ 0 ι 0 = θ 1 ι 1 = Id ( i.e. the sections of them, taking primitives for the corresponding differential operators). Then Li − s 1 ,..., − s r = ( θ t 1 +1 ι 1 . . . θ t r +1 ι 1 )1 Ω and Li s 1 ,..., s r = ( ι s 1 − 1 ι 1 . . . ι s r − 1 ι 1 )1 Ω . 0 0 0 0

Noncommutative, co-commutative bialgebras X and Y are ordered, respectively, by x 1 > x 0 and y 1 > y 2 > . . . . L ynX , { S l } l ∈L ynX : pure transcendence bases of 1 ( C � X � , ⊔ , 1 X ∗ ) , ⊔ L ynY , { Σ l } l ∈L ynY : pure transcendence bases of 2 ( C � Y � , , 1 Y ∗ ) , { P l } l ∈L ynX , { Π l } l ∈L ynY : homogeneous (graded) bases of Lie algebras of primitive elements, for respectively ∆ ⊔ ⊔ , ∆ , ◮ in the concatenation-shuffle bialgebra ( C � X � , conc , ∆ ⊔ ⊔ , 1 X ∗ , e ), ց � � e S l ⊗ P l D X := w ⊗ w = (MRS-factorization). w ∈ X ∗ l ∈L ynX ◮ in the concatenation-stuffle bialgebra ( C � Y � , conc , ∆ , 1 Y ∗ , e ), ց ( − extended � � e Σ l ⊗ Π l D Y := w ⊗ w = MRS-factorization) . w ∈ Y ∗ l ∈L ynY 1 For x , y ∈ X , u , v ∈ X ∗ , u ⊔ ⊔ 1 X ∗ = 1 X ∗ ⊔ ⊔ u = u and ⊔ yv = x ( u ⊔ ⊔ yv ) + y ( xu ⊔ ⊔ v ), or equivalently xu ⊔ ⊔ ( x ) = x ⊗ 1 X ∗ + 1 X ∗ ⊗ x ( i.e. letters are primitive, for ∆ ⊔ ∆ ⊔ ⊔ ). 2 For y i , y j ∈ Y , u , v ∈ Y ∗ , u 1 Y ∗ = 1 Y ∗ u = u and y i u y j v = y i ( u y j v ) + y j ( y i u v ) + y i + j ( u v ), or equivalently ( y i ) = y i ⊗ 1 Y ∗ + 1 Y ∗ ⊗ y i + � ∆ k + l = i y k ⊗ y k ( i.e. y 1 is primitive, for ∆ ).

First structures of polylogarithms and harmonic sums 0 ( z ) := log k ( z ) / k !, { Li w } w ∈ X ∗ is C -linearly 1. Completed with Li x k independent. Hence, the following morphism of algebras is injective Li • : ( C � X � , ⊔ , 1 X ∗ ) → ( C { Li w } w ∈ X ∗ , ., 1) , u �→ Li u . ⊔ Thus, { Li l } l ∈L ynX (resp. { Li S l } l ∈L ynX ) are algebraically independent. 2. The following morphism of algebras is injective H • : ( C � Y � , , 1 Y ∗ ) → ( C { H w } w ∈ Y ∗ , ., 1) , u �→ H u . Hence, { H w } w ∈ Y ∗ is C -linearly independent. It follows that, { H l } l ∈L ynY (resp. { H Σ l } l ∈L ynY ) are algebraically independent. ( Q 1 X ∗ ⊕ x 0 Q � X � x 1 , ⊔ , 1 X ∗ ) ⊔ 3. ζ : , 1 Y ∗ ) ։ ( Z , ., 1) such that, for ( Q 1 Y ∗ ⊕ ( Y − { y 1 } ) Q � Y � , ⊔ l 2 ) = ζ (( π Y l 1 ) any l 1 , l 2 ∈ L ynX − X , ζ ( l 1 ⊔ ( π Y l 2 )) = ζ ( l 1 ) ζ ( l 2 ). 4. There exists, at least, an associative law of algebra ⊤ , in Q � Y 0 � , (not dualizable) such that the following morphism is onto Li − ( Q � Y 0 � , ⊤ ) ( Q { Li − w �→ Li − : → w } w ∈ Y ∗ 0 , . ) , w , • and ker Li − • = Q { w − w ⊤ 1 Y ∗ 0 | w ∈ Y ∗ 0 } . Moreover, if ⊤ ′ : Q � Y 0 � × Q � Y 0 � → Q � Y 0 � is a law such that Li − • is a morphism for ⊤ ′ and (1 Y ∗ 0 ⊤ ′ Q � Y 0 � ) ∩ ker( Li − • ) = { 0 } then ⊤ ′ = g ◦ ⊤ , where g ∈ GL ( Q � Y 0 � ) such that Li − • ◦ g = Li − • .

SINGULAR AND ASYMPTOTIC EXPANSIONS

Noncommutative series and first Abel like theorem ց ց � � � e Li Sl P l , e ζ ( S l ) P l , L := Li w w = ( Li • ⊗ Id ) D X = Z ⊔ ⊔ := w ∈ X ∗ l ∈L ynX l ∈L ynX − X ց ց � � � e H Σ l Π l , e ζ (Σ l )Π l . H := H w w = ( H • ⊗ Id ) D Y = Z := w ∈ Y ∗ l ∈L ynY l ∈L ynY −{ y 1 } L is solution of ( DE ) satisfying L ( z ) ∼ 0 e x 0 log( z ) . One has L ( z ) ∼ 1 e − x 1 log(1 − z ) Z ⊔ ⊔ . Theorem (HNM, 2005) k ≥ 1 H yk ( n )( − y 1 ) k / k H ( n ) = π Y Z ⊔ z → 1 e y 1 log(1 − z ) π Y L ( z ) = lim � lim n →∞ e ⊔ . For w ∈ X ∗ x 1 , there exists a i , b i , j ∈ Z and α i , β i , j , γ π Y w ∈ Z [ γ ] such that | w | � � a i log i (1 − z ) + � Z ⊔ b i , j (1 − z ) j log i (1 − z ) , Li w ( z ) ≍ ⊔ | w � + z → 1 i =1 i , j ∈ N + ( w ) log i ( n ) � α i log i ( n ) + γ π Y w + � H π Y w ( n ) ≍ β i , j . n j n → + ∞ i =1 i , j ∈ N + Let Z γ := � w ∈ Y ∗ γ w w . Then Z γ is group-like, for ∆ . By the extended MRS-factorization, one has Z γ = e γ y 1 Z and then, by the Abel like = B ′ ( y 1 ) π Y Z ⊔ theorem, one deduces ( Z γ = B ( y 1 ) π Y Z ⊔ ⊔ ⇔ Z ⊔ ), k ≥ 2 ζ ( k )( − y 1 ) k / k and B ′ ( y 1 ) = e − � k ≥ 2 ζ ( k )( − y 1 ) k / k . where B ( y 1 ) = e γ y 1 − �