DRINFEL’D ASSOCIATOR AND RELATIONS OF SOME SPECIAL FUNCTIONS

DRINFEL’D ASSOCIATOR AND RELATIONS OF SOME SPECIAL FUNCTIONS. Dong Thap University Journal of Science, Vol 11, No 5, 2022, 19 28 19 DRINFEL’D ASSOCIATOR AND RELATIONS OF SOME SPECIAL FUNCTIONS Bui Van Chien University of Sciences, Hue University Email bvchienhu.

Dong Thap University Journal of Science, Vol 11, No 5, 2022, 19-28 DRINFEL’D ASSOCIATOR AND RELATIONS OF SOME SPECIAL FUNCTIONS Bui Van Chien University of Sciences, Hue University Email: bvchien@hueuni.edu.vn Article history Received: 07/5/2022; Received in revised form: 08/7/2022; Accepted: 27/7/2022 Abstract We observe the differential equation dG( z) / dz  ( x0 / z  x1 / (1  z))G( z) in the space of power series of noncommutative indeterminates x0 , x1 , where the coefficients of G( z ) are holomorphic functions [(,0)  (1, )] Researches on this equation in some conditions on the simply connected domain turn out different solutions which admit Drinfel'd associator as a bridge In this paper, we review representations of these solutions by generating series of some special functions such as multiple harmonic sums, multiple polylogarithms and polyzetas Thereby, relations in explicit forms or asymptotic expansions of these special functions from the bridge equations are deduced by identifying local coordinates Keywords: Drinfel'd associator, multiple harmonic sums, multiple polylogarithms, polyzetas, special functions LIÊN HỢP DRINFEL’D VÀ QUAN HỆ CỦA MỘT SỐ HÀM ĐẶC BIỆT Bùi Văn Chiến Trường Đại học Khoa học, Đại học Huế Email: bvchien@hueuni.edu.vn Lịch sử báo Ngày nhận: 07/5/2022; Ngày nhận chỉnh sửa: 08/7/2022; Ngày duyệt đăng: 27/7/2022 Tóm tắt Chúng tơi quan sát phương trình vi phân dG( z) / dz  ( x0 / z  x1 / (1  z))G( z) không gian chuỗi lũy thừa phần tử khơng giao hốn x0 , x1 , hệ số G( z ) hàm chỉnh [(,0)  (1, )] Những nghiên cứu xung quanh phương trình hình miền đơn liên số điều kiện khác cho ta nghiệm khác liên hợp Drinfiel'd cầu nối chúng Trong báo này, tổng quan lại việc biểu diễn trường hợp nghiệm thông qua hàm sinh hàm đặt biệt tổng điều hòa bội, hàm polylogarit bội chuỗi zeta bội Từ phương trình cầu nối, rút quan hệ dạng tường minh khai triển tiệm cận các hàm đặc biệt cách đồng tọa độ địa phương Từ khóa: Liên hợp Drinfel'd, tổng điều hòa bội, hàm polylogarit bội, chuỗi zeta bội, tổng điều hòa bội DOI: https://doi.org/10.52714/dthu.11.5.2022.976 Cite: Bui Van Chien (2022) Drinfel’d associator and relations of some special functions Dong Thap University Journal of Science, 11(5), 19-28 19 Natural Sciences issue Introduction n * Let and ( n * : {( z1 ,, zn )  ∣ zi  z j for i  j} n ) denotes the ring of holomorphic functions over the universal covering of n * denoted by Using n n * , : {tij }1i  j n as an alphabet, Knizhnik and Zamolodchikov (1984) defined a noncommutative first order differential equation acting in the ring ( n * ) n  ,  1i  j  n tij 2i (1.1) G ( z1 , z2 )  exp(  (t12 dz  t23 dz )G( z) 2i z 1 z : (1.2) ( )t12 , t23  , where [(,0)  (1, )] (1.3) t13 t , x1 : 23 , equation (1.2) 2i 2i can be rewritten as follows x x dG( z )  (  )G( z ), dz z 1 z (1.4) and more shortly dG( z )  (0 ( z) x0  1 ( z) x1 )G( z) by using the two differential forms dz dz 0 ( z ) : and 1 ( z ) : z 1 z Chen series, of 0 and 1 along a path z on , defined by (Cartier, 1987): 20 z :  wX * z z0 xik  X * , as follows z  zz ( w)   i ( z1 ) z0 zk 1 z0 i ( zk ), k * The series Cz0 ( w) w  ( ) X , e is group-like (Ree, 1958), z Lz0 z  Cz0 z (1.7) In (Drinfel'd, 1990), Drinfel'd is essentially interested in solutions of (1.4) over the interval (0;1) and, using the involution z  z , he stated (1.4) admits a unique solution G0 (resp G1 ) satisfying asymptotic forms (1.8) Moreover, G0 and G1 are group-like series then there is a unique group-like series  KZ  X , Drinfel'd series (so-called G0  G1 KZ (1.6) (1.9) After that, via a regularization based on representation of the chord diagram algebras Le Tu Quoc Thang and Murakami (1996) expressed the divergent coefficients of  KZ as linear combinations of Multiple-Zeta-Value (or polyzetas) defined for each composition (s1 ,, sr )  r1 , s1  2, as follows  ( s1 ,, sr ) : (1.5) The resolution of (1.4) uses the so-called Cz0 any Drinfel'd associator), such that By taking x0 : z0 for G0 ( z ) z 0 z x0 and G1 ( z) z 1 (1  z ) x1 In the case n  , the equation is appplied in the ring w  xi1 defined, which implies that there exists a primitive series Lz0 z such that For example, with n  , one has  {t12 } and a solution of the equation dG( z)  2G( z) , t where 2  12 d log( z1  z2 ) , is 2i dG( z )  is  (1X )  d log( zi  z j ) t12 log( z1  z2 )) 2i  ( z1  z2 )t12 / 2i  ( 2* )  zz (w)  ( ) z z0 dG( z )  n ( z )G( z ), where n : where X * denotes the free monoid generated by the alphabet X (equipping the empty word as the neutral element) and, for a subdivision ( z0 , z1 , zk , z ) of z0 z and the coefficient  n nrsr s1 n1  nr 1 (1.10) In other words, these polyzetas can be reduced by the limit at z  of multiple polylogarithms or at N   of multiple harmonic sums, respectively defined on each multi-index (s1 ,, sr )  r1 , r  , and z  , z  1, n  , as follows Dong Thap University Journal of Science, Vol 11, No 5, 2022, 19-28 Li s1 ,, sr ( z ) : H s1 ,, sr (n) : z n1  s1 sr , n1  nr 1 n1 nr n  n nrsr s1 n1  nr 1 (1.11) conc  w  (1.12) Moreover, the multiple harmonic sums can be viewed as coefficients of generating series of the multiple polylogarithm for each multi-index (1  z )1 Li s1 ,, sr ( z )   H s1 ,, sr (n) z n The associative unital concatenation, denoted by and its co-law which is denoted  by conc and defined for any as follows (1.13) Algebras of shuffle and quasi-shuffle products The above special functions are compatible with shuffle and quasi-shuffle structures In order to represent these properties more clearly, we correspond each multi-index (s1 ,, sr )  r1 , r  to words generated by the two alphabets X  {x0 , x1} and Y  { yk }k 1 as follows ( s1 ,, sr )  x0s1 1 x1  x0sr 1 x1  X * x1 X Y ys1  ysr  Y * , (2.1) Where and respectively denote the free monoids of words generated by the alphabets and with the empty words 1X * and 1Y * (sometime use in common) as the neutral elements This section reviews two structures of shuffle and quasishuffle algebras compatible with the special functions introduced above 2.1 Bi-algebras in duality By taking formal sums of words, we can extend the monoids and to the -modules, X and Y , which become bidenoted by algebras with respect to the following product and co-product: (2.2) The associative commutative and unital shuffle product defined, for any x, y  X and u, v  X * , by the recursion uш1X *  1X * шu  u, n 1 In this work, we review a method to construct relations of the special functions by following equation (1.9) The generating series of the special functions are group-like series to review simultaneously the essential steps to furnish G0 and  KZ which follows related equations in asymptotic expansion forms and then an equation bridging the algebraic structures of converging polyzetas  u  v; uv  w xuшyv  x(uшyv)  y ( xuшv), (2.3) or equivalently, by its coproduct (which is a morphism for concatenations) defined, for each letter x  X , as follows ш x  1X *  x  x  1X * (2.4) According to the Radford theorem (Radford, 1979), LynX forms a pure transcendence basis of the -shuffle algebras, graded in length of word, ( X , ш,1X * ) (Reutenauer, 1993) Similarly, the Y is also equipped with the -module associative commutative and unital stuffle product defined, for u, v, w Y * and yi , y j  Y , by wж1Y *  1Y * ж w  w, yi u ж y j v  yi (u ж y j v)  y j ( yi u жv)  yi  j (u жv) It can be dualized according to yk  Y   yk  : yk  1Y *  1Y *  yk  which is also a algebra (  y y i  j k morphism and the i j -stuffle Y , Ж,1Y * ) admits the set of Lyndon words, denoted by LynY , as a pure transcendence basis (Hoang, 2013; Bui Van Chien et al., 2015) This algebra is graded in weight defined by taking sum of all index of letters in a word For example, the weight of the word w  ys1  ysr is s1  sr Note that, the stuffle product defined here just acts on the monoid generated by alphabet Y but the shuffle product can be applied for any alphabet 21 Natural Sciences issue We will use as a general alphabet used for shuffle product and A as a field extension of , SA Y be the sets of formal series extended from A X AY  respectively Then and Definition 2.1 Let A i S is said to be a group-like series if and ш S  S  S (resp only if  S1 *  and  ж S  S  S ) * where the counit being here e( P)  P |1 Moreover, this algebra is graded and admits a Poincaré-BirkhoffWitt basis (Reutenauer, 1993) {Pw }w * which is expanded from the homogeneous basis {Pl }lLyn of the Lie algebra of concatenation product, denoted by Its graded dual basis is denoted by ieA {Sw }w ii S is said to be a primitive series if and only if ш S   S  S 1 (resp ж S  1Y  S  S 1Y ) * * * [ x; y]  x  y  y  x The following results are standard facts from works by Ree (Ree, 1958) (see also (Bui Van Chien et al., 2015; Reutenauer, 1993) Proposition 2.1 i The Lie bracket of two primitive elements is primitive ii Let S  A Y ) Then S is (resp A primitive, for  ж (resp  conc and ш ), if and only * if, for any u, v  Y *Y (resp ), we get S | u жv  (resp S | uv  and S | uшv  ) Proposition 2.2 Let S  A Y (resp A Then the following assertions are equivalent i S is a ж -character (resp character) ) and ш - ii S is group-like, for  ж (resp  conc and ш ) iii log S is primitive, for  ж (resp  conc and ш ) Corollary 2.1 Let S  A Y * admitting the pure transcendence basis {Sl }lLyn of the A -shuffle algebra * The Lie bracket in an algebra is defined for some algebra with the product () as usual (resp A ) Then the following assertions are equivalent i S an infinitesimal ж -character (resp and ш -character) ii S is primitive, for  ж (resp  conc and ш ) 2.2 Factorization in bi-algebras Due to Cartier-Quillin-Milnor-Moore (Cartier, 1987) theorem (CQMM theorem), it is well known that the enveloping algebra ( ie ) is 22 isomorphic to the (connected, graded and cocommutative) bialgebra ш ( )  ( A , conc,1 , ш , e), In the case when A is a -algebra, we also have the following factorization of the diagonal series, (Reutenauer, 1993) (here all tensor products are over A ) D :  ww   e w Sl  Pl lLyn * (2.5) and (still in the case A is a -algebra) dual bases of homogeneous polynomials {Pw }w * and {Sw }w * can be constructed recursively as follows  Px  x,for x  ,   Pl  [ Pl1 , Pl2 ],(l )  (l1 , l2 ),  ik ik i1 i1  Pw  Pl1  Plk , LF ( w)  l1 lk , (2.6) where LF ( w) denotes the Lyndon factorization of the word which is rewritten a word as a product of decreasing Lyndon words    S x  x, x  ,   Sl  ySl  , l  yl   Lyn  ,  шik шi1  S  Sl1 шшSlk , LF ( w)  l i1 l ik (2.7) k  w i1 !ik ! The graded dual of  ш ( )  (A ш ( ) is , ш,1 * , conc , ) We get another connected, graded and cocommutative bialgebra which, in case A is a algebra, is isomorphic to the enveloping algebra of the Lie algebra of its primitive elements, ж (Y )  ( AY , conc,1Y * ,  ж , )  (Prim( ж (Y ))), (2.8) Dong Thap University Journal of Science, Vol 11, No 5, 2022, 19-28 bases of homogeneous polynomials { w }wY * and where Prim( (Y ))  Im(1 )  span A{1 ( w) | w Y } * ж and  is defined, for any w  Y * , by (Hoang, 2013; Bui Van Chien et al., 2015) { w }wY * Now, let { w }wY * be the linear basis, expanded by decreasing Poincaré-Birkhoff-Witt (PBW for short) after any basis {l }lLynY of Prim( ж (Y )) be its dual basis which contains the pure transcendence basis {l }lLynY of the A -stuffle algebra One also has the factorization of the diagonal series DY , on ж (Y ) , which reads (Bui Van Chien et al., 2013) DY :  ww   e l  l , (2.9) lLynY wY * where the last expression takes product of exponential in decreasing of Lyndon words We are now in the position to state the following Theorem 2.1 (Hoang, 2013) Let A be a endomorphism of algebras -algebra, then the  : ( AY , conc,1Y )  ( AY , conc,1Y ) * (Y )  (Prim( constructed directly and In  ys   ( ys )for ys  Y ,   l  [ l1 ,  l2 ]for l  LynY Y ,,  ik ik i1 i1  w   l1  lk for w  l1 lk , (2.10)   yk  y k ,  l   ys1 l1ln (*)1      ys  s l l , i n  i  i ! (*)2  lж1 i1 ж  ж lжk ik   w  i1 !ik !  (2.11) (*) , the sum is taken over all {k1 ,, ki }  {1,, k} and l1   ln such that * * ( ys1 ,, ysr ) ( ysk ,, ysk , l1 ,, ln ) , where  i denotes the transitive closure of the relation on standard sequences, denoted by  (Bui Van Chien et al., 2013; Reutenauer, 1993) Drinfel’d associator with special functions 3.1 Relations among multiple polylogarithms and multiple harmonic sums * mapping yk to 1 ( yk ) , is an automorphism of AY  realizing an isomorphism of bialgebras between ш (Y ) and ж be recursively by ( w) ( 1)k 1   ( w)  w   w|u1жжuk u1uk (2.8) k k 2 u1 ,,uk Y  homogeneous in weight, and let { w }wY * can ж By correspondence (3.1) and the properties of the special functions, we can define the following (morphisms) are injective Li• : (  X , ш,1X * )  ( {Li w }wX * ,.,1), (Y ))) x0n log n ( z ) / n!, x0s1 1 x1  x0sr 1 x1 In particular, it can be easily checked that the following diagram commutes Li x s11x x sr 1x 1 and H• : ( Y , ш,1Y * )  ( {H w }wY * ,.,1), ys1  ysr Hence, the bases { w }wY * and { w }wY * of (Prim( ж (Y ))) are images by 1 and by the adjoint mapping of its inverse,  v H ys ys  H s1 ,, sr (3.2) r Hence, the families {Li w }wX * and {H w }wY * are linearly independent of {Pw }wY * and Now, using DX and DY , the graphs of Li• {Sw }wY * , respectively Algorithmically, the dual and H • are given as follows (Hoang, 2013; Bui Van Chien et al., 2015) 1 23 Natural Sciences issue L : (Li•  Id) X   e Li Sl Pl e Li Sl Pl  Y ( Zш )  lim e y log(1 z ) Y (L( z )) , z 1  H yk ( n )(  y1 )k / k lLynX and L reg   n  , lLynX l  x0 , x1 e H : (H •  Id) DY   lim e k 1 H l  l Hence, the coefficients of any word w in Z ш and Z ж respectively represent the finite parts (denoted by f p ) of asymptotic expansion of multiple polylogarithm and multiple harmonic sum in the scales of comparison , lLynY and H reg  e H l  l {(1  z )a logb ((1  z ) 1 )}a lLynY l  y1 series in regularization taking convergent words, the words are coded by convergent multi-index of polyzetas Moreover, we set Zш : Lreg (1) and Zш : Hreg () z (3.3) , L, Lreg , and then Z ш (resp ш ) Moreover, L is also a solution of (1.4) Theorem 2.1 (Cristian and Hoang, 2009; Bui Van Chien et al., 2015) lim L( z )e z L( z0 )  L( z ),  x0 log( z ) z 0  1, f p.n H w (n)  Z Ж | w Example 2.1 (Cristian and Hoang, 2009) In convergence case, Li 2,1 ( z )   (3)  (1  z )log(1  z )  (1  z ) 1 L( z ) z 1 e x1 log(1 z ) Zш (1  z )log (1  z ) / (1  z ) ( log (1  z )  log(1  z )) /  , H 2,1 (n)   (3)  (log(n)    ) / n  log(n) / 2n  , one has Li1,2 ( z )   2 (3)   (2)log(1  z ) This means that for x0  A / 2i and x1   B / 2i , L corresponds to G0 expected by Drindfel'd and Z ш corresponds to  KZ , 2(1  z )log(1  z )  (1  z )log (1  z ) (1  z ) (log (1  z )  log(1  z )) /  , H1,2 (n)   (2)  2 (3)   (2)log(n) ( (2)  2) / 2n  , Via Newton-Girard identity type, we also get (Cristian and Hoang, 2009; Bui Van Chien et al., 2015)  H yk ( n )(  y1 )k / k k H ( n ) y  e  yk k1 k 0 since numerically,  (2)  0.949481711114981524545564, then one has f.p.z 1 Li1,2 ( z )   2 (3), f.p.n H1,2 (n)   (2)  2 (3) and then H(n) z  ( H yk (n) y ) Y ( Zш ) k 0 k It follows that Theorem 2.2 (Cristian and Hoang, 2009; Bui Van Chien et al., 2015) Moreover, the relations among the multiple polylogarithms indexed by basis {Sl }lLynX follow Li S x ( z )  log( z ), Li S x ( z )   log(1  z ), Li S x x ( z )   log( z )log(1  z )  Li S x x (1  z ) 01 01  ( S x0 x1 ), 24 In divergence case z 1 and , b f.p.z 1 Li 2,1 ( z)  f.p.n H2,1 (n)   (2,1)   (3) lim e x1 log(1 z ) L( z )  Zш L( z) z 0 e x0 log( z ) ,{na H1b (n)}a f p.z 1 Li w ( z )  Zш | w , H, H reg , and then Z ш ) are grouplike, for ш (resp Cz0 , b This means that We note that L reg and H reg are generating As for Cz0 H(n) Dong Thap University Journal of Science, Vol 11, No 5, 2022, 19-28 Li S x0 x12 Theorem 2.3 (Hoang, 2013; Bui Van Chien et al., 2015) ( z )  log(1  z ) log( z )  log(1  z )Li S x x (1  z ) Z  B( y1 ) Y Zш 01 Li S (1  z )   ( S x2 x ) or equivalently by simplification x0 x1 Z ж  B( y1 ) Y Zш  log( z ) ( S x0 x1 ) Using the correspondences given in (3.4), let us consider then the following -algebra of convergent polyzetas, being algebraically generated by { (l )}lLynX  X (resp { ( Sl )}lLynX  X ), or equivalently, { (l )}lLynY { y1 } by (resp { (l )}lLynY { y1 } ):  span { ( w)}wY * y1Y * (3.5) For any k  let  k y1  k : span { ( w)}wx X * x s1 , , s r  1, s1  ks r  k  k s1 ! s r ! (  ) k  ( x0 [(  x1 ) ш X w]) i0 i!  k i s1 (  (2) ) s2 (    (k ) ) sr i ( b i, j , k ) ( ,  (2), 2 (3), ) , j 1 where k   , w Y  Bell polynomials and bn,k (t1 ,, tk ) are Example 2.2 (Cristian and Hoang, 2009) (3.6) ( w)  k Now, considering the third and last noncommutative generating series of polyzetas (Cristian and Hoang, 2009; Bui Van Chien et al., 2015) Z   w w, (3.7) wY *  w  f.p.n Hw (n) on  w  f.p.n Hw (n),{n log (n)}a a b ,b the scale For any w Y * y1Y * , one has  w   (w) and  y1   (Euler's constant) The series Z  is group-like, for  Ж Then (Hoang, 2013; Bui Van Chien et al., 2015)  lLynY e (l ) l  e y1 Zш (3.8) { y1 } Moreover, introducing the following ordinary generating seriesi B( y1 ) : exp( y1   (k ) k 2  (  3 (2)  2 (3)) 54   (7)   (3) (5)   (2) , 175   (2)3   ( (2) (5)   (3) (2) 35 19 4 (7))   (6, 2)   (2) 35   (2) (3)  4 (3) (5)  1,1  (   (2)),  1,1,1  span { ( w)}w(Y { y })Y * Z  e y1 ( 1)   With the correspondences given in (3.13), we get w k where (3.12) Identifying the coefficients in these identities, we get y1 w : span { ( w)}wx X * x k (3.11) ( y1 )k ), k ( y1 )k B( y1 ) : exp(  (k ) ), k k 2 we obtain the following bridge equation (3.9)  1,7  1,1,6 3.2 Relations of polyzetas As the limits limLi s ( z )  lim H s (n)   ( s) z 1 n  for any convergent multi-indexii s , polyzetas inherits properties both of multiple polylogarithms and multiple harmonic sums We can define polyzetas as a morphism of shuffle and quasishuffle products from ( 1X *  x0 Xx1 , ш,1X * ) or ( 1Y *  (Y { y1}) Y , ж,1Y * ) onto -algebra, denoted by , algebraically generated by the convergent polyzetas { (l )}lLynX  X (Bui Van Chien et al., 2015) It (3.10) can be extended as characters  :( X ,,1X * )  ( ,.,1), 25 Natural Sciences issue  , • : ( Y ,,1Y * )  ( ,.,1) : [  ш ( w)  f p.z 1 Li w ( z ),  ж ( Y w)  f p.n H w (n),  irr  [ ( )]  irr ( ) ] Moreover, the following sub ideals of ker  as follows ,b ( X )]   [{ ( p)}p such that, for any w  X * , one has the finite part corresponding the scales a b a b and {(1  z) log (1  z)}a ,b , {n H1 (n)}a ,b {na logb (n)}a  irr RY : (span {Ql }lLynY { y1 } RX : (span {Ql }lLynX X , ж ,1Y * ), , ш,1X * ) are generated by the polynomials {Ql } lLyn , l{ y1 , x0 , x1 } Y  Yw  f p.n HY l (n) It follows that,  ш ( x0 )   log(1) and the finite parts, corresponding the scales {(1  z)a logb (1  z)}a ,b ,{na H1b (n)}a ,b , a b {n log (n)}a homogeneous in weight such that the following assertions are equivalent: i Ql  , ii l  l (resp Sl  Sl ), , as follows ,b iii l   irr (Y ) (resp Sl   irr ( X ) )  ш ( x1 )   f p.z 1 log(1  z ),  ш ( y1 )   f p.n H1 (n),  y    f p.n H1 (n) Any polynomial Ql (  ) is led by  l (resp Sl ), being transcendent over the sub algebra and the following convergent polyzetas, (resp Sl  U l ) being homogeneous of weight p  (l ) and belonging to [ l  LynX  X ,  irr  ш (l )   ж ( Y l )    l   (l ), [ Y  ш ( Sl )   ж ( Y Sl )    Y Sl   ( Sl ), span {l }lLynY l In (Cristian and Hoang, 2009), polynomial relations among { (l )}lLynX  X (or { (l )}lLynY { y1 } ), are obtained using the double shuffle relations The identification of local coordinates in Z  B( y1 ) Y Zш , leads to a family of algebraic 2 irr ( ) ( X ) of p irr  p irr  ( )   irr ( ) ( )  p irr  p irr ( )   irr ( ) ( ) p2 such that the following restriction is bijective 26  irr  R  span { y1 } ( ) which  R  span  irr follows ( ) For any w  x0 X * x1 (resp Y { y1})Y * ), by the Radford's theorem (Reutenauer, 1993), one has  (w)  [ irr ( )] Hence, for any P  [{Sl }lLynX X ] (resp [{l }lLynY { y1} ] such that P  ker   R , one gets, by linearity,  ( P)  [  irr ( )]  ( )] , the polymorphism  is bijective on [ irr and then Q  It follows that and their inverse image by a section of  ( ) X  Q  R  [ irr ( )] Since Next, let R  ker  then  (Q)  Moreover, restricted p2 2 irr ( ))] In other terms, l  Ql   l i.e Sl  Ql  Ul (resp  ж (l )   ш ( X l )   l   (l ),  ж (l )   ш ( X l )      (l )  irr p irr span {Sl }lLynX l  LynY  { y1}, generators ( )], and l   l Proposition 2.3 (Hoang, 2013; Bui Van Chien et al., 2015) [{Sl }lLynX [{l }lLynY X ]  RX  [  irr ( X )], { y1 } ]  RY  [  irr (Y )] Dong Thap University Journal of Science, Vol 11, No 5, 2022, 19-28 Via CQMM theorem and by duality, one deduces then Corollary 2.2 ( ie ( ie X)  X { y1 })  Y X      ie     Y      ie     { Pl } lLynX  (X ) Sl  irr    ,     {l } lLynX  (Y ) l  irr    ,     Now, irr l let Q  ker  , Corollary 2.4 (Hoang, 2013; Bui Van Chien et al., 2015)  1  k Q  Q1  Q2 with Q1  R and Q2  [ Thus, Q  R Q1  R and then Then  irr ( )] Corollary 2.3 (Hoang, 2013; Bui Van Chien et al., 2015) [{ ( p)}p  ( )]  (3.14) k 2 irr Q1 *   1X *  x0 Xx1 / ker  Hence, as an ideal generated by homogeneous in weight polynomials, ker  is graded and so is : where X (resp Y ) is a Lie ideal generated by {Pl }lLynX :S   ( X ) (resp {l }lLynY :   (Y ) ) l Y / ker   1Y *  (Y  { y1})  Im and R  ker  Now, let    ( P), where P  and P  ker  , homogeneous in weight Since, for any p and n  1, one has  p n pn then each monomial  n , for n  1, is of different weight Thus  could not satisfy  n  an1 n1  0, with an1 , Corollary 2.5 (Hoang, 2013; Bui Van Chien et  al., 2015) Any s  irr ( ) is homogeneous in weight then  ( s) is transcendent over irr On the other hand, one also has Example 2.3 Polynomials relations on local coordinates (Bui Van Chien et al., 2015) Due to the bridge equation (3.12), we obtain Table Table Polynomial relations of polyzetas on transcendence bases 27 Natural Sciences issue Example 2.4 (Bui Van Chien et al 2015) List of irreducible polyzetas up to weight 12 for each transcendence basis: 12 irr  ( S x x ), ( X )  { (S x0 x1 ),  ( S x x x x ), 1  ( S x x ),  ( S x x ),  ( S x x x x ),  ( S x x ), 1  ( S x x ),  ( S x x x x ),  ( S x x x x ),  ( S x x x x )} 10 12 irr 1 1 1 (Y )  { ( y2 ),  ( y3 ),  ( y5 ),  ( y7 ),  ( y y ),  ( y ),  ( y y ),  ( y ),  ( y y ), 9 11  ( y y ),  ( y y )} Conclusion We reviewed a method to reduce relations of the special functions indexed by transcendence bases of shuffle and quasi-shuffle algebras due to the Drinfel'd associator Starting from the research of Knizhnik-Zamolodchikov about a form of a differential equation, a bridge equation is constructed, and it can be applied to the case of the generating series of the special functions Relations in form of asymptotic expansions or explicit representations hold by the identification of local coordinates of the bridge equation References Cartier, P (1987) Jacobienne généralisées, monodromie unipotente et intégrales intérées Paris: Séminaire BOURBAKI Chien, B V and Duchamp, G H E and Hoang, N M V (2013) Schützenberger's factorization on the (completed) Hopf algebra of q-stuffle product JP Journal of Algebra, Number Theory and Applications, 30, 191-215 Chien, B V and Duchamp, G H E and Hoang, N M V (2015) Structure of Polyzetas and Explicit Representation on Transcendence Bases of Shuffle and Stuffle Algebras P Symposium on Symbolic and Algebraic Computation, 40, 93-100 i ii For any k  1, log B( y1 ) | y1k  f.p.n  s  (s1,, sr )  28 r H l 1 yl Chien B V.; 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