-deformed shuffle bialgebras and renormalization V.C. B` ui, - PowerPoint PPT Presentation

ϕ -deformed shuffle bialgebras and renormalization V.C. B` ui, G.H.E. Duchamp, Hoang Ngoc Minh, Q.H. Ngˆ o Paths to, from and in renormalization February, 8th-12th 2016, Potsdam

Plan 1. Introduction 1.1 Renormalization of (all) divergent polyzetas ? 1.2 Structures of harmonic sums and of polylogarithms at positive indices 1.3 Shifts and derivations of shuffle-conc bialgebra 2. Background on combinatorics of ϕ -deformed of shuffle bialgebras 2.1 ( A � Y � , ., 1 Y ∗ , ∆ x , ǫ Y ) and its deformations 2.2 ϕ -shuffle products 2.3 ϕ -extended Sch¨ utzenberger’s factorization in quasi-shuffle-conc bialgebras 3. Abel like theorems for noncommutative generating series 3.1 Global renormalization of polyzetas at positive indices 3.2 Global renormalization of polyzetas at negative indices 3.3 Bi-integro-differential algebra C{ Li w } w ∈ X ∗ 4. Some consequences on structure of polyzetas 4.1 Homogenous polynomials relations among local coordinates 4.2 Noetherian rewriting system and irreducible local coordinates 4.3 Examples irreducible local coordinates by computer

INTRODUCTION

Renormalization of (all) divergent polyzetas ? For z ∈ C , | z | < 1, let ( s 1 , . . . , s r ) ∈ C r , r ∈ N + , the polylogarithm is well defined z n 1 � Li s 1 ,..., s r ( z ) := . n s 1 1 . . . n s r r n 1 >...> n r > 0 Then the Taylor expansion of (1 − z ) − 1 Li s 1 ,..., s r ( z ) is given by Li s 1 ,..., s r ( z ) � H s 1 ,..., s r ( N ) z N , = 1 − z N ≥ 0 where the coefficient H s 1 ,..., s r ( N ) is a harmonic sum which can be expressed as follows 1 � H s 1 ,..., s r ( N ) := . n s 1 1 . . . n s r r N ≥ n 1 >...> n r > 0 m � For any m = 1 , .., r , if ℜ ( s i ) > 1 then, after a theorem by Abel, one i =1 obtains 1 the polyzeta as 1 � z → 1 Li s 1 ,..., s r ( z ) = lim lim N →∞ H s 1 ,..., s r ( N ) = ζ ( s 1 , . . . , s r ) := n s 1 1 . . . n s r r n 1 >...> n r > 0 else ??? 1 see the talk of Guo.

Encoding multi-indices by words Let X ∗ and Y ∗ be the free monoids (admitting 1 X ∗ and 1 Y ∗ as units) generated respectively by X = { x 0 , x 1 } and Y = { y k } k ≥ 1 . Here, we suppose that 2 ( s 1 , . . . , s r ) ∈ N r + . π Y v = x s 1 − 1 x 1 . . . x s r − 1 s = ( s 1 , . . . , s r ) ↔ u = y s 1 . . . y s r ⇋ π X x 1 . 0 0 For s 1 > 1, the associated words in x 0 X ∗ x 1 or ( Y − { y 1 } ) Y ∗ are said to be convergent. For r ≥ k ≥ 1, a divergent word is of the following form 1 x s k +1 − 1 x 1 . . . x s r − 1 ( { 1 } k , s k +1 , . . . , s r ) ↔ y k π Y x k 1 y s k +1 . . . y s r ⇋ π X x 1 . 0 0 Let Y ∗ 0 be the free monoid generated by Y 0 = Y ∪ { y 0 } with 1 Y ∗ 0 as unit. ( s 1 , . . . , s r ) ∈ N r ↔ y s 1 . . . y s r ∈ Y ∗ 0 . The length and the weight of w = y s 1 . . . y s r ∈ Y ∗ or Y ∗ 0 (resp. w = x s 1 . . . x s r ∈ X ∗ ) are respectively | w | = r , for Y ∗ or Y ∗ 0 , (resp. X ∗ ) and ( w ) = s 1 + . . . + s r , for Y ∗ and Y ∗ 0 . Let L ynY 0 , L ynY and L ynX denote the sets of Lyndon words respectively over Y 0 , Y and X , totally ordered by x 0 < x 1 and y 0 > y 1 > y 2 , · · · . 2 see the minicourses of Ebrahimi-Fard and Singer.

Indexing polylogarithms and harmonic sums by words Let ( s 1 , . . . , s r ) ∈ N r + . Then Li s 1 ,..., s r ( z ) = Li y s 1 ... y sr ( z ) = x 1 ( z ) , Li x s 1 − 1 x 1 ... x sr − 1 0 0 H s 1 ,..., s r ( N ) = H y s 1 ... y sr ( N ) = H x s 1 − 1 x 1 ( N ) , x 1 ... x sr − 1 0 0 ζ ( x s 1 − 1 x 1 . . . x s r − 1 ζ ( s 1 , . . . , s r ) = ζ ( y s 1 . . . y s r ) = x 1 ) . 0 0 Let Z denote the Q -algebra generated by convergent polyzetas. Let ( s 1 , . . . , s r ) ∈ N r . Then 3 Li − � n s 1 1 . . . n s r r z n 1 , y s 1 ... y sr ( z ) := Li − s 1 ,..., − s r ( z ) = n 1 >...> n r > 0 � n s 1 H − 1 . . . n s r := = r , H − s 1 ,..., − s r y s 1 ... y sr N ≥ n 1 >...> n r > 0 � n s 1 1 . . . n s r ζ − ( y s 1 . . . y s r ) := ζ ( − s 1 , . . . , − s r ) ↔ r . n 1 >...> n r > 0 3 Previous works on renormalization of ζ ( − s 1 , . . . , − s r ) : ◮ D. Manchon, S. Paycha , Nested sums of symbols and renormalised multiple zeta functions, 2010. ◮ L. Guo, B. Zhang , Differential Birkhoff decomposition and the renormalization of multiple zeta values , 2012. ◮ H. Furusho, Y. Komori, K. Matsumoto, H. Tsumura , Desingularization of multiple zeta-functions of generalized Hurwitz-Lerch type , 2014.

Harmonic sums as monomial quasi-symmetric functions Let N , r ∈ N , r > 0 and let s = ( s 1 , . . . , s r ) ∈ ( N + ) ∗ be the multi-index associated to the word w = y s 1 . . . y s r ∈ Y ∗ . Using the correspondence ( N + ) ∗ ∋ ( s 1 , . . . , s r ) = s w = y s 1 . . . y s r ∈ Y ∗ , ↔ the monomial quasi-symmetric functions, on t = { t i } i ≥ 1 , are defined by � t s 1 n 1 . . . t s k M 1 Y ∗ ( t ) = M ∅ ( t ) = 1 and M w ( t ) = M s ( t ) = n k . n 1 >...> n k > 0 For any u , v ∈ X ∗ , one has ( Knutson ’s inner product, 1973) M u v ( t ) = M u ( t ) M v ( t ) . H s 1 ,..., s r ( N ) (resp. H − s 1 ,..., − s r ( N )) is obtained then by specializing the indeterminates t = { t i } i ≥ 1 in the monomial quasi-symmetric function M s ( t ) = M w ( t ) respectively as follows ( Hoffman , 1997) t i = 1 / i (resp. t i = i ) and ∀ i > N , t i = 0 . Hence, Q { H w } w ∈ Y ∗ ∼ = Q � Y � and ( HNM , 2003) ( Q { H w } w ∈ Y ∗ , × ) ∼ ) ∼ = ( Q � Y � , = ( Q [ L ynY ] , ) .

Polylogarithms as iterated path integrals The iterated integral, associated to w ∈ X ∗ , along the path z 0 � z and over the differential forms ω 0 ( z ) = dz / z and ω 1 ( z ) = dz / (1 − z ), is defined, on any appropriate simply connected domain Ω, as follows 4 . 1 Ω if w = 1 X ∗ , � � z α z z 0 ( w ) = ω i 1 ( t ) α t x i 1 ∈ X , u ∈ X ∗ . z 0 ( u ) if w = x i 1 u , z 0 For any u , v ∈ X ∗ , one has ( Chen ’s lemma, 1954) α z z 0 ( u x v ) = α z z 0 ( u ) α z z 0 ( v ) . { Li w } w ∈ X ∗ x 1 are obtained then as iterated integrals ∀ w ∈ X ∗ x 1 , Li w ( z ) = α z 0 ( w ) . Setting Li x 0 ( z ) = log z = α z 1 ( x 0 ), one can use iterated integrals, with z 0 = 0, to calculate the other values, via a theorem by Radford (1956) because Q { Li w } w ∈ X ∗ ∼ = Q � X � and ( HNM, Petitot, Hoeven , 1998) ( Q { Li w } w ∈ X ∗ , × ) ∼ = ( Q � X � , x ) ∼ = ( Q [ L ynX ] , x ) (or more generally Deneufchˆ atel, Duchamp, HNM, Solomon , 2011). 4 see the talk of Panzer.

Derivations and shifts in shuffle algebras Definition Let S ∈ C � � X � � (resp. C � X � ) and P ∈ Q � X � (resp. C � � X � � ). The right (resp. left ) residual of P by S , is P ⊳ S (resp. S ⊲ P ) defined by 5 : ∀ w ∈ X ∗ , � P ⊳ S | w � = � P | Sw � (resp. � S ⊲ P | w � = � P | wS � ) . In particular, for any x , y ∈ X and w ∈ X ∗ , x ⊲ ( wy ) = ( yw ) ⊳ x = δ y x w . These residuals (or shifts) are associative and commute with each other : S ⊲ ( P ⊳ R ) = ( S ⊲ P ) ⊳ R , P ⊳ ( RS ) = ( P ⊳ R ) ⊳ S , ( RS ) ⊲ P = R ⊲ ( S ⊲ P ) . Proposition (derivations and automorphisms) Let P ∈ C � X � (resp. C � � X � � ) and T ∈ C � � X � � (resp. C � X � ) such that ∆ x ( T ) = 1 ⊗ T + T ⊗ 1 . Then ◮ P �→ P ⊳ T and P �→ T ⊲ P are derivations of ( C � X � , x , 1 X ∗ ) (resp. ( C � � X � � , x , 1 X ∗ ) ). ◮ P �→ P ⊳ exp( tT ) and P �→ exp( tT ) ⊲ P are one-parameter groups of automorphisms of ( C � X � , x , 1 X ∗ ) (resp. ( C � � X � � , x , 1 X ∗ ) ). 5 These actions are the shifts of functions in harmonic analysis.

COMBINATORICS OF ϕ -SHUFFLE-CONC BIALGEBRAS

( A � Y � , ., 1 Y ∗ , ∆ x , ǫ Y ) and its deformations A : commutative and associative algebra with unit over Q . Let A � Y � and A � � Y � � denote the sets of polynomials and of formal power series over Y , with coefficients in A , equipped with the concatenation. They are also endowed with the ϕ -shuffle defined recursively by ∀ w ∈ Y ∗ ,  w ϕ 1 Y ∗ = 1 Y ∗ ϕ w = w ,  ∀ y i , y j ∈ Y , ∀ u , v ∈ Y ∗ , y i u ϕ y j v = y i ( u ϕ y j v ) + y j ( y i u ϕ v ) + ϕ ( y i , y j )( u ϕ v ) ,  where ϕ is an arbitrary mapping defined by its structure constants � γ k ϕ : Y × Y − → AY , ( y i , y j ) �− → i , j y k . k ∈ I ⊂ N + It is said to be dualizable if there exists ∆ ϕ : A � Y � → A � Y � ⊗ A � Y � � ∗ � such that the dual mapping A � Y � ⊗ A � Y � → A � � Y � � restricts to ϕ . Theorem (Duchamp, Enjalbert, HNM, Tollu, 2014) 1. The law ϕ is associative (resp. commutative) if and only if the linear extension ϕ : AY ⊗ AY − → AY is so. 2. Let γ z x , y := � ϕ ( x , y ) | z � be the structure constants of ϕ , then ϕ is dualizable if and only if ( γ z x , y ) x , y , z ∈ Y has the following property ( ∀ z ∈ Y )(# { ( x , y ) ∈ Y 2 | γ z x , y � = 0 } < + ∞ ) .