Structure of polyzetas and the algorithms to express them on algebraic bases on words Gérard H.E. DUCHAMP , HOANG NGOC MINH, Van Chiên BUI LIPN - Université Paris 13 Journées Nationales de Calcul Formel, 3 − 7 Novembre 2014 1/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
Outline Introduction Hopf algebras of noncommutative polynomials Global regularizations of polyzetas Polynomial relations among polyzetas 2/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
Outline Introduction Hopf algebras of noncommutative polynomials Global regularizations of polyzetas Polynomial relations among polyzetas 3/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
Introduction Definition For each s = ( s 1 , . . . , s r ) ∈ ( N ∗ ) r , s 1 > 1, the polyzetas (multiple zeta values (MZVs)) is defined by the following convergent series 1 � ζ ( s ) = ζ ( s 1 , . . . , s r ) := n s 1 1 . . . n s r r n 1 >...> n r > 0 4/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
Introduction Definition For each s = ( s 1 , . . . , s r ) ∈ ( N ∗ ) r , s 1 > 1, the polyzetas (multiple zeta values (MZVs)) is defined by the following convergent series 1 � ζ ( s ) = ζ ( s 1 , . . . , s r ) := n s 1 1 . . . n s r r n 1 >...> n r > 0 Example ∞ ∞ 1 1 � � ζ ( 2 ) = n 2 , ζ ( 3 ) = n 3 , n = 1 n = 1 1 � ζ ( 2 , 3 ) = . n 2 1 n 3 2 n 1 > n 2 > 0 4/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
Introduction Definition For each s = ( s 1 , . . . , s r ) ∈ ( N ∗ ) r , s 1 > 1, the polyzetas (multiple zeta values (MZVs)) is defined by the following convergent series 1 � ζ ( s ) = ζ ( s 1 , . . . , s r ) := n s 1 1 . . . n s r r n 1 >...> n r > 0 4/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
Introduction Definition For each s = ( s 1 , . . . , s r ) ∈ ( N ∗ ) r , s 1 > 1, the polyzetas (multiple zeta values (MZVs)) is defined by the following convergent series 1 � ζ ( s ) = ζ ( s 1 , . . . , s r ) := n s 1 1 . . . n s r r n 1 >...> n r > 0 Theorem (comparison formula) [9] Z γ = Γ( y 1 + 1 ) π Y ( Z ⊔ ⊔ ) (1) B ′ ( y 1 ) π Y ( Z ⊔ ⇐ ⇒ Z = ⊔ ) (2) 4/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
Outline Introduction Hopf algebras of noncommutative polynomials Global regularizations of polyzetas Polynomial relations among polyzetas 5/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
Algebraic structures on noncommutative polynomial On the alphabet X = { x 0 , x 1 } [4] Hopf algebras in duality ( Q � X � , . , 1 X ∗ , ∆ ⊔ ⊔ , s ) ⇄ ( Q � X � , ⊔ ⊔ , 1 X ∗ , ∆ conc , s ) , One constructed the PBW basis ( P w ) w ∈ X ∗ of the freely associated algebra Q � X � and the transcendent basis ( S l ) l ∈L ynX of the algebra ( Q � X � , ⊔ ⊔ , 1 X ∗ ) . 6/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
Algebraic structures on noncommutative polynomial On the alphabet X = { x 0 , x 1 } [4] Hopf algebras in duality ( Q � X � , . , 1 X ∗ , ∆ ⊔ ⊔ , s ) ⇄ ( Q � X � , ⊔ ⊔ , 1 X ∗ , ∆ conc , s ) , One constructed the PBW basis ( P w ) w ∈ X ∗ of the freely associated algebra Q � X � and the transcendent basis ( S l ) l ∈L ynX of the algebra ( Q � X � , ⊔ ⊔ , 1 X ∗ ) . On the alphabet Y = ( y s ) s ∈ N ∗ [2] Hopf algebras in duality , s ′ ) ⇄ ( Q � Y � , , 1 Y ∗ , ∆ conc , s ′ ) , ( Q � Y � , . , 1 Y ∗ , ∆ We also constructed the PBW basis (Π w ) w ∈ Y ∗ of the freely associated algebra Q � Y � and the transcendent basis (Σ l ) l ∈L ynY of the algebra ( Q � Y � , , 1 Y ∗ ) . 6/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
Constructing linear relation between C := Q ⊕ Q � X � x 1 and Q � Y � , view as the graded modules ( 1 / 4 ) 7/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
Constructing linear relation between C := Q ⊕ Q � X � x 1 and Q � Y � , view as the graded modules ( 1 / 4 ) We now consider homogeneous polynomials by length with respect to alphabet X and by weight with respect to alphabet Y . We view the following graded vector spaces � � C := Q ⊕ Q � X � x 1 = span Q ( X n ) ≃ Q � Y � = span Q ( Y n ) n ≥ 0 n ≥ 0 where X n , Y n being corresponding the sets of words with length and weight n . Example X 0 := { 1 } , Y 0 := { 1 } X 1 := { x 1 } , Y 1 := { y 1 } Y 2 := { y 2 , y 2 X 2 := { x 0 x 1 , x 1 x 1 } , 1 } { x 2 0 x 1 , x 0 x 2 1 , x 1 x 0 x 1 , x 3 Y 3 := { y 3 , y 2 y 1 , y 1 y 2 , y 3 X 3 := 1 } , 1 } . . . 7/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
Constructing linear relation between C := Q ⊕ Q � X � x 1 and Q � Y � , view as the graded modules ( 2 / 4 ) Definition Let us define the linear isomorphism π Y : C − → Q � Y � x s 1 − 1 x 1 . . . x s r − 1 x 1 �− → y s 1 . . . y s r 0 0 and π X be its inverse. Its extension over Q � X � , be still denoted by π Y , satisfying π Y ( p ) = 0 for any p ∈ Q � X � x 0 . Example π Y ( x 0 ) = 0 ,π Y ( x 1 ) = y 1 π Y ( x 0 x 1 ) = y 2 π Y ( 2 x 0 x 1 − x 1 x 0 − 1 2 x 1 x 0 x 1 ) = 2 y 2 − 1 2 y 1 y 2 8/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
Constructing linear relation between C := Q ⊕ Q � X � x 1 and Q � Y � , view as the graded modules ( 3 / 4 ) Lemma For any w ∈ X ∗ x 1 , we call P ′ w , S ′ w to be corresponding P w , S w restricted on C . Then, ( P ′ w ) w ∈ X n , ( S ′ w ) w ∈ X n are a pair of bases in duality of the span Q ( X n ) . Example ⇒ P ′ = S ′ P x 1 = S x 1 = x 1 ∈ C x 1 = x 1 , x 1 ⇒ P ′ P x 0 x 1 = x 0 x 1 − x 1 x 0 ∈ C / = x 0 x 1 , x 0 x 1 ⇒ S ′ S x 0 x 1 = x 0 x 1 ∈ C = x 0 x 1 , x 0 x 1 = x 0 x 2 1 − 2 x 1 x 0 x 1 + x 2 ⇒ P ′ = x 0 x 2 P x 0 x 2 1 x 0 ∈ C / 1 − 2 x 1 x 0 x 1 , x 0 x 2 1 1 = x 0 x 2 ⇒ S ′ S x 0 x 1 x 1 ∈ C = x 0 x 1 1 x 0 x 1 Remark: For any w ∈ X ∗ , we have π Y ( P ′ w ) = π Y ( P w ) and π Y ( S ′ w ) = π Y ( S w ) . 9/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
Constructing linear relation between C := Q ⊕ Q � X � x 1 and Q � Y � , view as the graded modules ( 4 / 4 ) For each n , we arrange the elements of X n and Y n in the increasing order respectively as u ( n ) < u ( n ) < . . . < u ( n ) 2 n − 1 , 1 2 v ( n ) < v ( n ) < . . . < v ( n ) 2 n − 1 ; and then establishing the matrix 1 2 representation of π Y in the two ordered bases as follow π Y P ′ π Y S ′ Π Σ u ( n ) v ( n ) u ( n ) v ( n ) 1 1 1 1 . . . . = M ( n ) = N ( n ) . . and . . . . . . . π Y P ′ π Y S ′ Π Σ v ( n ) v ( n ) u ( n ) u ( n ) 2 n − 1 2 n − 1 2 n − 1 2 n − 1 10/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
� � � Constructing linear relation between C := Q ⊕ Q � X � x 1 and Q � Y � , view as the graded modules ( 4 / 4 ) For each n , we arrange the elements of X n and Y n in the increasing order respectively as u ( n ) < u ( n ) < . . . < u ( n ) 2 n − 1 , 1 2 v ( n ) < v ( n ) < . . . < v ( n ) 2 n − 1 ; and then establishing the matrix 1 2 representation of π Y in the two ordered bases as follow π Y P ′ π Y S ′ Π Σ u ( n ) v ( n ) u ( n ) v ( n ) 1 1 1 1 . . . . = M ( n ) = N ( n ) . . and . . . . . . . π Y P ′ π Y S ′ Π Σ v ( n ) v ( n ) u ( n ) u ( n ) 2 n − 1 2 n − 1 2 n − 1 2 n − 1 By the duality, we have N ( n ) := ( t ( M ( n ) )) − 1 . We can see more clearly by the following diagram π Y ( span Q ( X n ) , ( P ′ ) 1 ≤ i ≤ 2 n − 1 ) ( span Q ( Y n ) , (Π ) 1 ≤ j ≤ 2 n − 1 ) v ( n ) u ( n ) M ( n ) i j duality duality π Y ( span Q ( X n ) , ( S ′ � ( span Q ( Y n ) , (Σ ) 1 ≤ i ≤ 2 n − 1 ) ) 1 ≤ j ≤ 2 n − 1 ) v ( n ) u ( n ) N ( n ) j i 10/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
Outline Introduction Hopf algebras of noncommutative polynomials Global regularizations of polyzetas Polynomial relations among polyzetas 11/22 BUI, DUCHAMP , HNM Structure of polyzetas and the expressing algorithms
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