Combinatorial study of colored Hurwitz polyzêtas

Combinatorial study of colored Hurwitz polyzêtas Jean-Yves Enjalbert, Hoang Ngoc Minh To cite this version: Jean-Yves Enjalbert, Hoang Ngoc Minh. Combinatorial study of colored Hurwitz polyzêtas. 2011. ฀hal-00704926฀ HAL Id: hal-00704926 https://hal.archives-ouvertes.fr/hal-00704926 Preprint submitted on 6 Jun 2012 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Combinatorial study of colored Hurwitz polyzêtas June 6, 2012 Jean-Yves Enjalbert1 , Hoang Ngoc Minh1,2 1 Universit Paris 13, Sorbonne Paris Cit, LIPN, CNRS(, UMR 7030), F-93430, Villetaneuse, France. 2 Université Lille II, 1 place Déliot, 59024 Lille, France Email adresses : [email protected], [email protected] abstract A combinatorial study discloses two surjective morphisms between generalized shuffle algebras and algebras generated by the colored Hurwitz polyzêtas. The combinatorial aspects of the products and co-products involved in these algebras will be examined. 1 Introduction P zêta function Classically, the Riemann zêta function is ζ(s) = n>0 n−s , the Hurwitz
P P is ζ(s; t) = n>0 (n − t)−s and the colored zêta function is ζ sq = n>0 q s n−s , where q is a root of unit. The three previous functions are defined over Z>0 but can be generalized over any composition (sequence of positive (s1 , . . . , sr ), P integers) s = −s 1 n . . . nr−sr , like, respectively, the Riemann polyzêta function ζ(s) = n1 >...>nr >0 1 P −s1 the Hurwitz polyzêta function ζ(s; t) = n1 >...>nr >0 (n1 − t1 ) . . . (nr − tr )−sr
P and the colored polyzêta function ζ qsi = n1 >...>nr >0 q i1 n1 . . . q ir nr n1−s1 . . . nr−sr , with q a root of unit and i = (i1 , . . . , ir ) a composition. These sums converge when s1 > 1. To study simultaneously these families of polyzêtas, the colored Hurwitz polyzêtas, for a composition s = (s1 , . . . , sr ) and a tuple of complex numbers ξ = (ξ1 , . . . , ξr ) and a tuple of parameters in ] − ∞; 1[, t = (t1 , . . . , tr ), are defined by [6] Di(Fξ,t ; s) = X n1 >...>nr ξ1n1 . . . ξrnr . (n1 − t1 )s1 . . . (nr − tr )sr >0 (1) Note that, for l = 1 . . . , r, the numbers ξl are not necessary roots of unity q il . We are working, in this note, with the condition 1 (E) ∀i, | i Y ξk | ≤ 1 and ti ∈] − ∞; 1[. k=1 Hence, Di(Fξ,t ; s) converges if s1 > 1. We note E the set of C-tuples verifying (E). These polyzêtas are obtained as special values of iterated integrals1 over singular differential 1-forms introduced in [10]. As iterated integrals, they are encoded by words or by non commutative formal power series [10] and are used to construct bases for asymptotic expanding [14] or symbolic integrating fuchian differential equations [11] exactly or approximatively [8]. The meromorphic continuation of the colored Hurwitz polyzêtas2 is already studied in [5, 6]. In our studies, we constructed an integral representation3 of colored Hurwitz polyzêtas and a distribution treating simultanously two singularities and our methods permit to make the meromorphic continuation commutatively over the variables s1 , . . . , sr [5, 6]. Moreover, [6] gives another way to obtain the meromorphic continuation thanks to translation equations [4]. Our methods give the structure of multi-poles [5] (Theorem 4.2) and two ways to calculate algorithmically the multi-residus4. In this note, in continuation with our previous works [10, 11, 12, 13, 5, 6], we are focusing on Hofp algebra, for a class of products as minusstuffle ( ), mulstuffle ( q ), . . . , and in particular for the new product duffle ( q ), obtained as “tensorial product” of q and the well known stuffle ( ), of symbolic representations of these polyzêtas (see Definition 2.1 and Proposition 2.1 bellow). 2 Combinatorial objects 2.1 Some products and their algebraic structures Let X be an encoding alphabet and the free monoid over X is denoted by X ∗ . The length of any word w ∈ X ∗ is denoted by |w| and the unit of X ∗ is denoted by 1X ∗ . For any unitary commutative algebra A, a formal power series S over X with coefficients P in A can be written as the infinite sum w∈X ∗ hS|wiw. The set of polynomials (resp. formal power series) over X with coefficients in A is denoted by AhXi (resp. AhhXii). The set of degree 1 monomials is AX = {ax/a ∈ A, x ∈ X}. Definition 2.1 We note P the set of products ⋆ over AhXi verifying the conditions : 1 They are presented as generalized Nielsen polylogarithms in [10] (Definition 2.3) and as generalized Lerch functions in [12] (Definition 3). 2 See also references and a discussion about meromorphic continuation of Riemann polyzêtas in [5]. 3 This integral representation is obtained by applying successively the polylogarithmic transform [10]. It is an application of non commutative convolution as shown in [9] (Section 2.4). Other integral representations can be also deduced easily by change of variables, for example t = zr and then r = e−u [5]. 4 Other meromorphic continuations can also be obtained by Mellin transform as already done in [17] or by classical estimation on the imaginary part [7] but these later work reccursively, depth by depth, and the commutativity of this process over the variables s1 , . . . , sr must be proved. Unfortunately, the structure of multi-poles as well as multi-residus are missing in both works [7, 17]. In [16], to make the meromorphic continuation (giving the expression of non positive integers multi-residus via a generalization of Bernoulli numbers – but not of all multi-residus) of the specialization at roots of unity of colored Hurwitz polyzêtas Di(Fξ,t ; s), the author bases on the integral representation, on the contours, of the multiple Hurwitz-Lerch which corresponds mutatis mutandis to the integral representation of generalized Lerch functions introduced earlier in [5] (Corollary 3.3). 2 (i) the map ⋆ : AhXi × AhXi → AhXi is bilinear, (ii) for any w ∈ X ∗ , 1X ∗ ⋆ w = w ⋆ 1X ∗ = w, (iii) for any a, b ∈ X and u, v ∈ X ∗ , au ⋆ bv = a(u ⋆ bv) + b(au ⋆ v) + [a, b](u ⋆ v), where [., .] : AX × AX → AX is a function verifying : (S1) ∀a ∈ AX, [a, 0] = 0 , (S2) ∀(a, b) ∈ (AX)2 , [a, b] = [b, a], (S3) ∀(a, b, c) ∈ (AX)3 , [[a, b], c] = [a, [b, c]]. Example 1 (see [18]) Product of interated integrals. The shuffle is a bilinear product such that : ∀w ∈ X ∗ w ⊔⊔ 1X ∗ = 1X ∗ ⊔⊔ w = w and 2 ∗2 ∀(a, b) ∈ X , ∀(u, v) ∈ X , au ⊔⊔ vb = a(u ⊔⊔ bv) + b(au ⊔⊔ v). For example, for any letter x0 , x and x′ in X, x0 x′ ⊔⊔ x20 x = x0 x′ x20 x + 2x20 x′ x0 x + 3x30 x′ x + 3x30 xx′ + x20 xx0 x′ . Example 2 (see [15]) Product of quasi-symmetric functions. Let X be an alphabet indexed by N. The stuffle is a bilinear product such that : ∀w ∈ X ∗ , w 1X ∗ = 1X ∗ w = w and ∀(xi , xj ) ∈ X 2 , ∀(u, v) ∈ X ∗ 2 , xi u xj v = xi (u xj v) + xj (xi u v) + xi+j (u v). In particular, with the alphabet Y = {y1 , y2 , y3 , . . .}, (y3 y1 ) y2 = y3 y1 y2 + y3 y2 y1 + y3 y3 + y2 y3 y1 + y5 y1 . Example 3 ([3]) Product of large multiple harmonic sums. Let X be an alphabet indexed by N. The minus-stuffle is a bilinear product such that : and ∀w ∈ X ∗ , w 1X ∗ = 1X ∗ w = w ∀(xi , xj ) ∈ X 2 , ∀(u, v) ∈ X ∗ 2 , xi u xj v = xi (u xj v) + xj (xi u v) − xi+j (u v). Example 4 ([6]) Product of colored sums. Let X be an alphabet indexed by a monoid (I, ×). The mulstuffle is a bilinear product such that : and ∀w ∈ X ∗ w q 1X ∗ = 1X ∗ q w = w ∀(xi , xj ) ∈ X 2 , ∀(u, v) ∈ X ∗ 2 , xi u q xj v = xi (u q xj v) + xj (xi u q v) + xi×j (u q v). For example, with X indexed by Q∗ , x 23 x−1 q x 12 = x 32 x−1 x 12 + x 23 x 12 x−1 + x 32 x −1 + x 21 x 23 x−1 + x 13 x−1 . 2 3 Remark 2.1 Thanks to the one-to-one correspondence (i1 , . . . , ir ) 7→ xi1 . . . xir between tuples of I and word over X, the calculus of x 23 x−1 q x 12 can be written as






2 q 1 = 2 , −1, 1 + 2 , 1 , −1 + 2 , −1 + 1 , 2 , −1 + 1 , −1 . 3 , −1 2 3 2 3 2 3 2 2 3 3 Example 5 ([6]) Product of colored Hurwitz polyzêtas. Let Y and E be two alphabets and consider the alphabet A = Y × E with the concatenation defined recursively by (y, e).(wY , wE ) = (ywY , ewE ) for any letters y ∈ Y , e ∈ E, and any word wY ∈ Y ∗ , wE ∈ E ∗ . The unit of the monoide A∗ is given by 1A∗ = (1Y ∗ , 1E ∗ ). If Y is indexed by N and E by a monoid (I, ×), the duffle is a bilinear product such that ∀w ∈ A∗ , w q 1A∗ = 1A∗ q w = w, ∀(yi , yj ) ∈ Y 2 , ∀(el , ek ) ∈ E 2 , ∀(u, v) ∈ A∗ 2 , (yi , el ).u q (yj , ek ).v = (yi , el ). (u q (yj , ek ).v) + (yj , ek ). ((yi , el ).u q v) + (yi+j , el×k ).(u q v). Proposition 2.1 The shuffle, the stuffle, the minus-stuffle and the mulstuffle are elements of P, with respectively, [xi , xj ] = 0, [xi , xj ] = xi+j , [xi , xj ] = −xi+j , [xi , xj ] = xi×j for any letters xi and xj of X. The duffle is in P, with [(yi , el ), (yj , ek )] = (yi+j , el×k ) for all yi , yj in Y , el , ek in E. Proposition 2.2 Let ⋆ ∈ P, then (AhXi, ⋆) is a commutative algebra. Proof. We just have to show the commutativity and the associativity of ⋆. To obtain w1 ⋆ w2 = w2 ⋆ w1 for all w1 , w2 in X ∗ , we use an induction on |w1 | + |w2 |. It is true when |w1 | + |w2 | ≤ 1 thanks to (i) since w1 or w2 is 1X ∗ . The equality (iii), the condition (S2) and the commutative of + give the induction. In the same way, an induction on |w1 | + |w2 | + |w3 | gives w1 ⋆ (w2 ⋆ w3 ) = (w1 ⋆ w2 ) ⋆ w3 thanks to (iii) and (S3). ✷ If we associate to each letter of X an integer number called weight, the weight of a word is the sum of the weight of its letters. In this case X is graduated. In [15], Hoffman works over X = X ∪ {0} with [., .] : X × X → X and call quasiproduct any product in P with the additional condition : (S4) Either [a, b] = 0 for all a, b in X; or the weight of [a, b] is the sum of the weight of a and the weight of b for all a, b in X. Example 6 1. The shuffle is a quasi-product. 2. Let X be an alphabet indexed by N and define the weight of xi , i ∈ N, by i . Then the stuffle is a quasi-product. Theorem 2.1 ([15]) If X is graduated and has a quasi-product ⋆, then (AhXi, ⋆) is a commutative graduated A-algebra.. (i) a comultiplication ∆ : AhXi → AhXi ⊗ AhXi, (ii) a counit ǫ : AhXi → A, ( X 1 if w = 1X ∗ ∗ by : ∀w ∈ X , ∆w = u ⊗ v and ǫ(w) = 0 otherwise. uv=w The coproduct ∆ is coassociative so (AhXi, ∆, ǫ) is a coalgebra. We can define 4 Lemma 2.1 For any w ∈ X ∗ and x ∈ X, (x ⊗ 1X ∗ )∆w + 1X ∗ ⊗ xw = ∆xw. X X Proof. ∀w ∈ X ∗ , ∀x ∈ X, ∆xw = u⊗v = xu′ ⊗ v + 1X ∗ ⊗ xw ′ uv=xw u v=w  X  so ∆xw = x ⊗ 1X ∗ u′ ⊗ v + 1X ∗ ⊗ xw = (x ⊗ 1X ∗ )∆w + 1X ∗ ⊗ xw. ✷ u′ v=w Proposition 2.3 If ⋆ ∈ P, then (AhXi, ⋆, ∆, ǫ) is a bialgebra. Remember that ⋆ acts over AhXi ⊗ AhXi by (u ⊗ v) ⋆ (u′ ⊗ v ′ ) = (u ⋆ u′ ) ⊗ (v ⋆ v ′ ). Proof. ǫ is obviously a ⋆-homomorphism. It still has to be show ∆(w1 ) ⋆ ∆(w2 ) = ∆(w1 ⋆ w2 ) over X ∗ . This equality is true if w1 or w2 is equal to 1X ∗ . Assume now that ∆(u)⋆∆(v) = ∆(u⋆v) for any word u and v such that |u|+|v| ≤ n, n ∈ N, and let w1 and w2 be in X ∗ with |w1 | + |w2 | = n + 1. We note w1 = au and ∗ w2 = bv,Pwith a and b two letters of X, u and v two words P of X . Thus, by definition, ∆w1 = u1 u2 =u au1 ⊗ u2 + 1X ∗ ⊗ au and ∆w2 = v1 v2 =v bv1 ⊗ v2 + 1X ∗ ⊗ bv. ∆(w1 ) ⋆ ∆(w2 ) X X = (au1 ⋆ bv1 ) ⊗ (u2 ⋆ v2 ) + (au1 ) ⊗ (u2 ⋆ bv) u1 u2 =u,v1 v2 =v + X u1 u2 =u (bv1 ) ⊗ (au ⋆ v2 ) + 1X ∗ ⊗ (au ⋆ bv) v1 v2 =v = X (a(u1 ⋆ bv1 ) ⊗ (u2 ⋆ v2 ) + b(au1 ⋆ v1 ) ⊗ (u2 ⋆ v2 ) u1 u2 =u,v1 v2 =v +([a, b](u1 ⋆ v1 )) ⊗ (u2 ⋆ v2 )) + X (au1 ) ⊗ (u2 ⋆ bv) u1 u2 =u + X (bv1 ) ⊗ (au ⋆ v2 ) + 1X ∗ ⊗ a(u ⋆ bv) v1 v2 =v +1X ∗ ⊗ b(au ⋆ v) + 1X ∗ ⊗ [a, b](u ⋆ v) X X = a(u1 ⋆ bv1 ) ⊗ (u2 ⋆ v2 ) + (au1 ) ⊗ (u2 ⋆ bv) u1 u2 =u,v1 v2 =v + X u1 u2 =u b(au1 ⋆ v1 ) ⊗ (u2 ⋆ v2 ) + u1 u2 =u,v1 v2 =v +[a, b] ⊗ 1X ∗ X X (bv1 ) ⊗ (au ⋆ v2 ) v1 v2 =v (u1 ⊗ u2 ) ⋆ (v1 ⊗ v2 ) u1 u2 =u v1 v2 =v +(1X ∗ ⊗ a(u ⋆ bv) + 1X ∗ ⊗ b(au ⋆ v) + 1X ∗ ⊗ [a, b](u ⋆ v)) = (a ⊗ 1X ∗ )(∆(u) ⋆ ∆(w2 )) + 1X ∗ ⊗ a(u ⋆ bv) + (b ⊗ 1X ∗ )(∆(w1 ) ⋆ ∆(v)) +1X ∗ ⊗ b(au ⋆ v) + ([a, b] ⊗ 1X ∗ )(∆(u) ⋆ ∆(v)) + 1X ∗ ⊗ [a, b](u ⋆ v). Using the induction hypothesis then the lemma 2.1 (since [a, b] ∈ AX) gives ∆(w1 ) ⋆ ∆(w2 ) = = = ∆(a(u ⋆ w2 )) + ∆(b(w1 ⋆ v)) + ∆([a, b](u ⋆ v)) ∆(a(u ⋆ w2 ) + b(w1 ⋆ v) + [a, b](u ⋆ v)) ∆(w1 ⋆ w2 ). ✷ 5 Remark 2.2 In particular, ∆ is a ⊔⊔ -homomorphism, a homomorphism. -homomorphism and a q - Let Cn be the set of positive integer sequences (i1 , . . . , ik ) such that i1 + . . . + ik = n. Theorem 2.2 Define a⋆ by, for all x1 , . . . , xn in X, a⋆ (x1 .X . . xn ) (−1)k x1 . . . xi1 ⋆ xi1 +1 . . . xi1 +i2 ⋆ . . . ⋆ xi1 +...+ik−1 +1 . . . xn = (i1 ,...,ik )∈Cn then, if ⋆ ∈ P, (AhXi, ⋆, ∆, ǫ, a⋆ ) is a Hopf algebra. Proof. With the applications : µ : A → AhXi m : AhXi ⊗ AhXi → AhXi and , λ 7→ λ 1X ∗ u⊗v 7→ u ⋆ v the antipode must verify m ◦ (a⋆ ⊗ Id) ◦ ∆ = µ ◦ ǫ, or, in equivalent terms X a⋆ (u) ⋆ v = hw|1X ∗ i1X ∗ . uv=w i.e. ( a⋆ (1X ∗ ) = 1X ∗ n ∀x ∈ X, a⋆ (x) = −x a⋆ (w) = − and, if w = x1 . . . xn with n ≥ 2, x1 , . . . , xn ∈ X, n−1 X a⋆ (x1 . . . xk ) ⋆ xk+1 . . . xn . k=1 An induction over the length n shows that a⋆ defined in theorem verifies these equalities, and, in the same way, a⋆ verifies m ◦ (Id ⊗ a⋆ ) ◦ ∆ = µ ◦ ǫ. ✷ Corollary 2.1 If ⋆ is ⊔⊔ or algebra. Moreover, for ⊔⊔ or or q or q , then this construction gives an Hopf , we obtain a graduated Hopf algebra. 2.2 Iterated integral Let us associate to each letter xi in X a 1-differential form ωi , defined in some connected open subset U of C. For all paths z0 z in U, the Chen iterated integral associated to w = xi1 · · · xik along z0 z, noted is defined recursively as follows Z z αz0 (w) = (2) ωi1 (z1 )αzz10 (xi2 · · · xik ) and αzz0 (1X ∗ ) = 1, z0 z verifying the rule of integration by parts [2] : αzz0 (u ⊔⊔ v) = αzz0 (u)αzz0 (v). We extended this definition over AhXi (resp. AhhXii) by X αzz0 (S) = hS|wiαzz0 (w). w∈X ∗ 6 (3) (4) 2.3 Shuffle relations 2.3.1 First encoding for colored Hurwitz polyzêtas Let ξ = (ξn ) be a sequence of complex numbers and T a family of parameters. Put X ′ an alphabet indexed over N∗ × CN × T and X = {x0 } ∪ X ′ . To each x in X we associate the differential form :  dz   si x = x0 ω0 (z) = z Qi (5) dz k=1 ξk   × t if x = xi,ξ,t with i > 1. Q ωi,ξ,t (z) = i z 1 − k=1 ξk z For any T -tuple t = (t1 , . . . , tr ) we associate the T -tuple t = (t1 , . . . , tr ) given by   t = t1 + t2 + . . . + t n , t1 = t1 − t2 ,      1    t2 = t2 − t3 ,  t2 = t2 + . . . + tn , in this way (6) .. ..     . .       tr = tr−1 − tr tr = tr We choose the sequence ξ and the family t such that the condition (E) is satisfied. Proposition 2.4 For any s = (s1 , . . . , sr ) with s1 > 1 if ξ = (ξ1 , . . . , ξr ) ∈ E r and t = (t1 , . . . , tr ) ∈ T r , then Di(Fξ,t ; s) = α10 (xs01 −1 x1,ξ,t1 . . . x0sr −1 xr,ξ,tr ). Proof. Since ωi,ξ,t (z) = i XY n>0 k=1 r XY ξkn r XY z n dz z n−tr z and then α (x ) = ξkn 0 r,ξ,tr 1+t z n − tr n>0 k=1 z n−tr . Hence, α10 (xs01 −1 x1,ξ,t1 . . . xs0r −1 xr,ξ,tr ) αz0 (x0sr −1 xr,ξ,tr ) = ξkn sr (n − t ) r n>0 k=1 Qj mj r X Y kj =1 ξkj gives s , and then, by change of vari(mj + . . . + mr − tj − . . . − tr ) j m1 ,...,mr >0 j=1 X ξ1n1 . . . ξrnr . ✷ ables, (n1 − t1 )s1 . . . (nr − tr )sr n >...>n >0 1 r Theorem 2.3 Let T be the group of parameters generated by hT ; +i, C be a subgroup of (C∗ , .) and A a sub-ring of C. Put C ′ = C N ∩ E and T ′ the set of finite tuple with elements in T . Then the A algebra generated by {Di(Fξ,t ; s)}ξ∈C ′ ,t∈T ′ is the A modulus generated by {Di(Fξ,t ; s)}ξ∈C ′ ,t∈T ′ . Proof. We have express the product Di(Fξ,t ; s) Di(Fξ′ ,t′ ; s′ ), with s = (s1 , . . . , sr ), s′ = (s′1 , . . . , s′r′ ), ξ, ξ ′ ∈ C ′ and t = (t1 , . . . , tr ), t′ = (t′1 , . . . , t′r ) ∈ T ′ , as linear combination of colored Hurwitz polyzêtas. This is an iterated integral associated s′ −1 to xs01 −1 x1,ξ,t1 . . . xs0r −1 xr,ξ,tr ⊔⊔ x01 s′ −1 x1,ξ′ ,t′ . . . x0r′ 1 7 xr′ ,ξ′ ,t′ which is a sum of r′ s′′ −1 terms of the form x01 s′′ −1 x1,ξ(1) ,t(1) . . . x0i s′′ −1 xji ,ξ(i) ,t(i) . . . x0r xjr′′ ,ξ(r′′ ) ,t(r′′ ) , with s′′i ∈ N, ξ (i) is ξ or ξ ′ and t(i) is tji or t′ji for all i; and r′′ = r + r′′ . Note that αz0 (xi,ξ,ti x0s−1 xj,ξ′ ,tj ) Z z XY Z i = ξkm z1m−ti −1 dz1 0 0 m>0 k=1 (ξ1 . . . ξi ) X = z1 m ξ1′ . . . ξj′
n dz2 ... z2 ′ ′ s m,n>0 (m + n − ti − tj )(n − tj ) s′′ −1 α10 x01 s′′ −1 x1,ξ(1) ,t(1) . . . x0i = r Y m1 ,...,mr′′ >0 i=1 X = n1 >...>nr zs+1 0 i XY dzs+1 z n+m , s′′ −1 (i) n−t′ −1 ξk′n zs+1 j n>0 k=1 xji ,ξ(i) ,t(i) . . . x0r ′′ X Z xjr′′ ,ξ(r′′ ) ,t(r′′ ) (i) (ξ1 . . . ξji )mi
′′ (mi + . . . + mr′′ − t(i) − . . . − t(r′′ ) )si ′′ r Y ξi′′ ni ′′ (ni − t′′i )si ′′ >0 i=1 (1) with ni = mi + . . . + mr′′ , t′′i = t(i) + . . . + t(r′′ ) for all i, so t′′ ∈ T ; ξ1′′ = ξ1 and ξi′′ = (i) (i) ξ1 ...ξj i (i−1) (i−1) ξ1 ...ξj i−1 for i > 1 so ξ ′′ ∈ C : we can express each term of the shuffle product as Di(Fξ′′ ,t′′ ; s′′ ). ✷ Note that the shuffle product over two words of X ∗ X ′ acts separately over (C ′ , .), (T ′ , +) and the convergent compositions. We can describe the situation with the shuffle algebra5 : Theorem 2.4 Let H be the Q-algebra generated by the colored Hurwitz polyzêtas. The ∗ map ζ : (Qh(x∗0 xi,ξ,t ) i, ⊔⊔ ) ։ (H, .), xs01 x1,ξ,t1 . . . xs0r xr,ξ,tr 7→ Di(Fξ,t ; s + 1) is a surjective algebra morphism. Example 7 Since Di(Fξ,t ; 3) = α10 (x20 x1,ξ,t ) and Di(Fξ′ ,t′ ; 2) = α10 (x0 x1,ξ′ ,t′ ) then Di(Fxi,t ; 3) Di(Fxi′ ,t′ ; 2) = α10 (x0 x1,ξ′ ,t′ ⊔⊔ x20 x1,ξ,t ). Example 1 with x = x1,ξ,t and x′ = x1,ξ′ ,t′ gives the expression of x0 x1,ξ′ ,t′ ⊔⊔ x20 x1,ξ,t . But the first term obtained is α10 (x0 x1,ξ′ ,t′ x20 x1,ξ,t ) Z 1 Z Z Z Z z2 dz1 z1 X ′ m m−t′ −1 dz3 z3 dz4 z4 X n n−t−1 = ξ z2 dz2 ξ z5 dz5 z3 0 z4 0 n>0 0 z1 0 m>0 0 m X ξ′ ξn = (m + n − t′ − t)2 (n − t)3 n,m>0 n X (ξ ′ ) 1 (ξ/ξ ′ )n2 = (n1 − t′ − t)2 (n2 − t)3 n >n >0 1 2 = Di(F(ξ,ξ/ξ′ );(t+t′ ,t) ; (2, 3)). 5 Working in Qh x∗0 xi,ξ,t
∗ i implies working in the graduated Hopf algebra (QhX ∗ i, ⊔⊔ , ∆, ǫ, a⊔⊔ ). 8 We can make similar calculus for the other terms and find : Di(Fξ,t ; 3) Di(Fξ′ ,t′ ; 2) = Di(F(ξ′ ,ξ/ξ′ );(t+t′ ,t) ; (2, 3)) + 2 Di(F(ξ′ ,ξ/ξ′ );(t+t′ ,t) ; (3, 2)) + 3 Di(F(ξ′ ,ξ/ξ′ );(t+t′ ,t) ; (4, 1)) + 3 Di(F(ξ,ξ′ /ξ);(t+t′ ,t′ ) ; (4, 1)) + Di(F(ξ,ξ′ /ξ);(t+t′ ,t′ ) ; (3, 2)). 2.3.2 Second encoding for colored Hurwitz polyzêtas For the Hurwitz polyzêtas, we can obtain an encoding indexed by a finite alphabet. Let the alphabet X = {x0 ; x1 } and associate to x0 the form ω0 (z) = z −1 dz and at x1 the form ω1 (z) = (1 − z)−1 dz. P For each x ∈ X and λ ∈ C, we note (λx)∗ = k≥0 (λx)k . Then, (see [10], [11]),
α10 xs01 −1 (t1 x0 )∗s1 x1 . . . xs0r −1 (tr x0 )∗sr x1 = ζ(s; t). Theorem 2.5 Let H′ be the Q-algebra generated by the Hurwitz polyzêtas and X the Q-algebra generated by (t1 x0 )∗s1 x1 . . . (tr x0 )∗sr xr . Then, ζ : (X , ⊔⊔ ) ։ (H′ , .) is a surjective morphism of algebras. Note that we can apply the idea of encoding of “simple” colored Hurwitz zetas functions (with depth one : r = 1). Let ξ = (ξn ) be a sequence of complex numbers in the unit ball B(0; 1) and T a family of parameters. Let X = {x0 , x1 , . . .} be a alphabet indexed by N. Associate to x0 the differential form ω0 (z) = z −1 dz and to xi , i ≥ 1, the differential form ωi (z) = ξi (1 − ξi z)−1 dz.  X ξn  s−1 i . Proposition 2.5 With this notation, α10 ((tx0 )∗ x0 ) (tx0 )∗ xi = (n − t)s n>0 Proof. Since X ξi dz0 = ξi (ξi z0 )n dz0 , we can write 1 − ξi z0 n≥0
αz0 (tx0 )k xi = tk Z 0 z dzk zk Z zk ... 0 Z z1 ξi 0 X (ξi z0 )n dz0 = n≥0 X n>0 tk ξin z n , nk+1 for z ∈ B(0; 1) and for k ∈ N. Thanks to the absolute convergence, αz0 ((tx0 )∗ xi ) = X ξ n z n X  t k X ξ n z n i i . = n n n −t n>0 n>0 k≥0 In the same way, if z ∈ B(0; 1) : ∀k ∈ N, so
X k ξin z n t αz0 (tx0 )k x0 (tx0 )∗ xi = , n − t nk+1 n>0 X ξn zn i αz0 ((tx0 )∗ x0 (tx0 )∗ xi ) = (n − t)2 n>0 9 and  X ξnzn  s−1 i . αz0 ((tx0 )∗ x0 ) (tx0 )∗ xi = (n − t)s n>0 ✷ Remark 2.3 Note that, with the same notation,   s−1 (t2 x0 )∗ x2 αz0 x1 ((t2 x0 )∗ x0 ) ξ2n ξ1m z n+m (n − t2 )s (m + n) n,m>0 X ξ n2 ξ n1 −n2 z n1 2 1 . n (n − t2 )s n >n >0 1 2 X = = 1 2 In other words, this encoding appears to be widespread only as couples of the type ξ = (1, 1, . . . , 1, ξr ) : with ξ1 = 1 and ω1 = (1 − z)−1 dz,   s −1 α10 x0s1 −1 (t1 x0 )∗s1 x1 . . . x0r−1 (tr−1 x0 )∗sr−1 xr−1 x0sr −1 (tr x0 )∗sr xr X ξrnr = . s (n1 − t1 ) 1 . . . (nr − tr )sr n >...>n 1 r 2.4 Duffle relations Let λ = (λn ) be a set of parameters, s = (s1 , . . . , sr ) a composition, ξ ∈ Cr . Then ∀n ∈ Z>0 , n Ms,ξ (λ) = r Y X ξini λsnii n and M(),() (λ) = 1. (7) n>n1 >...>nr >0 i=1 We can export the duffle over the tuples s = (s1 , . . . , sr ) ∈ Zr>0 and ξ ∈ Cr with : (s, ξ) q ((), 1) = ((), 1) q (s, ξ) = (s, ξ) and (s1 , s; ξ1 , ξ) q (r1 , r; ρ1 , ρ) = (s1 ; ξ1 ). ((s; , ξ) q (r1 , r; ρ1 , ρ)) + (r1 ; ρ1 ). ((s1 , s; ρ1 , ξ) +(s1 + r1 ; ξ1 ρ1 ). ((s; ξ) q (r; ρ)) q (r; ρ)) (8) Proposition 2.6 Let s = (s1 , . . . , sl ) and r = (r1 , . . . , rk ) be two compositions, ξ ∈ Cl , ρ ∈ Ck . Then ∀n ∈ N, n n n q (r,ρ) (λ). Ms,ξ (λ) Mr,ρ (λ) = M(s,ξ) Proof. Put the compositions s′ = (s2 , . . . , sl ), r′ = (r2 , . . . , rk ), the tuples of complex numbers ξ ′ = (ξ2 , . . . , ξl ) and ρ′ = (ρ2 , . . . , ρk ), then n n Ms,ξ (λ) Mr,ρ (λ) X ′ ′ ξ1n1 λsn11 Msn′1,ξ′ (λ) ρn1 1 λrn1′ 1 Mrn′ ,ρ1 ′ (λ) = = n>n1 ,n>n′1 X ξ1n1 λsn11 n>n1 n1 Msn′1,ξ′ (λ) Mr,ρ (λ) + X n>n′ 1 10 ′ ′ ′ n1 ρn1 1 λrn1′ 1 Ms,ξ (λ) Mrn′ ,ρ1 ′ (λ) + X s1 +r1 Msm′ ,ξ′ (λ) Mrm′ ,ρ′ (λ). (ξ1 ρ1 )m λm n>m A recurrence ended the demonstration. ✷ Theorem 2.6 Let s = (s1 , . . . , sl ) and r = (r1 , . . . , rk ) be two compositions, ξ a ltuple and ρ a k-tuple of E, t = (t, . . . , t) a l-tuple and t′ = (t, . . . , t) a k-tuple, both formed by the same parameter t diagonally. Then Di(Fξ,t ; s) Di(Fξ′ ,t′ ; s′ ) = Di(Fξ′′ ,(t,...,t) ; s′′ ), with (s′′ ; ξ ′′ ) = (s; ξ) Proof. With λn = q (s′ ; ξ ′ ). 1 for all n ∈ N, n−t n (λ) = Ms,ξ X r Y n>n1 >...>nr i=1 ξini . So (ni − t)si n lim Ms,ξ (λ) = Di(Fξ,t ; s) and taking the limit of Proposition 2.6 gives the result. n→∞ ✷ Example 8 The use of examples 2 and 4 gives Di(F( 23 ,−1),t ; (3, 1)) Di(F( 12 ),(t) ; (2)) = Di(F( 23 ,−1, 21 ),(t,t,t) ; (3, 1, 2)) + Di(F( 23 , 21 ,−1),(t,t,t) ; (3, 2, 1)) + Di(F( 23 ,− 12 ),t ; (3, 3)) + Di(F( 21 , 23 ,−1),(t,t,t) ; (2, 3, 1)) + Di(F( 31 ,−1),t ; (5, 1)) Remark 2.4 Extend the duffle product to triplets (s, t, ξ) ∈ ∪r∈N∗ Nr × {t}r × Cr by (s1 , s; t, t; ξ1 , ξ) q (r1 , r; t, t′ ; ρ1 , ρ) = (s1 ; t; ξ1 ). ((s; t; ξ) q (r1 , r; t, t′ ; ρ1 , ρ)) + (r1 ; t; ρ1 ). ((s1 , s; t, t; ρ1 , ξ) q (r; t′ ; ρ)) + (s1 + r1 ; t; ξ1 ρ1 ). ((s; t; ξ) q (r; t′ ; ρ)) , and define the function F over I = ∪r∈N∗ Nr × {t}r × Cr by F (s, t, ξ) = Di(Fξ,t ; s). Then, by Theorem 2.6, the function F : (I, q ) → (C, .) is morphism of algebras. References [1] E. Abe.– Hopf algebra, Cambridge, 1980. [2] K.T. Chen.– Iterated path integrals, Bull. Amer. Math. Soc., vol 83, 1977, pp. 831-879. 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