On Drinfel'd associators

TL;DR

This paper explores Drinfel'd associators, providing new interpretations and tools related to noncommutative evolution equations, with a focus on their role in braid group representations and asymptotic behaviors.

Contribution

It offers a novel interpretation of Drinfel'd associators and introduces new tools involving noncommutative evolution equations.

Findings

New interpretation of Drinfel'd associators

Development of tools for noncommutative evolution equations

Discussion of asymptotic phenomena related to associators

Abstract

In 1986, in order to study the linear representations of the braid group $B\_n$coming from the monodromy of the Knizhnik-Zamolodchikov differential equations,Drinfel'd introduced a class of formal power series $\Phi$on noncommutative variables. These formal series can be considered as a class of associators. We here give an interpretation of them as well as some new tools over Noncommutative Evolution Equations. Asymptotic phenomena are also discussed.

Equations

$$(DE)$$

$${\zeta }_r:{\mathcal{H}}_r\to \mathbb{R},$$

$${\mathrm{Li}}_{s_1,\dots ,s_k}(z)=\sum _{n_1>\dots >n_k>0}\frac{z^{n_1}}{n_1^{s_1}\dots n_k^{s_k}},{\mathrm{H}}_{s_1,\dots ,s_k}(N)=\sum _{n_1>\dots >n_k>0}^N\frac{1}{n_1^{s_1}\dots n_k^{s_k}}.$$

$$\underset{z\to 1}{\lim }{\mathrm{Li}}_{s_1,\dots ,s_k}(z)=\underset{n\to \infty }{\lim }{\mathrm{H}}_{s_1,\dots ,s_k}(n)=:{\zeta }_r(s_1,\dots ,s_k)$$

$$(1-z{)}^{-1}{\mathrm{Li}}_{s_1,\dots ,s_k}(z)=\sum _{n\ge 1}{\mathrm{H}}_{s_1,\dots ,s_k}(n)z^n,$$

$$\displaystyle\mathrm{Prim}({\mathcal{H}}_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;})$$

$${\pi }_1(w)$$

$$\displaystyle\mathrm{H}^{-}_{\bullet}:({\mathbb{Q}}\langle Y_{0}\rangle,\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;,1_{Y_{0}^{*}})\longrightarrow({\mathbb{Q}}\{\mathrm{H}^{-}_{w}\}_{w\in Y_{0}^{*}},\times,1),$$

$${\mathrm{Li}}_{\bullet }^{-}:(\mathbb{Q}⟨Y_0⟩,\top ,1_{Y_0^{\ast }})⟶(\mathbb{Q}\{{\mathrm{Li}}_w^{-}{\}}_{w\in Y_0^{\ast }},\times ,1),$$

$$\displaystyle\mathrm{H}_{\bullet}:({\mathbb{Q}}\langle Y\rangle,\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;,1_{Y^{*}})\longrightarrow({\mathbb{Q}}\{\mathrm{H}_{w}\}_{w\in Y^{*}},\times,1),$$

$${\mathrm{Li}}_{\bullet }:(\mathbb{Q}⟨X⟩,{}_{{}^{\bigsqcup \bigsqcup }},1_{X^{\ast }})⟶(\mathbb{Q}\{{\mathrm{Li}}_w{\}}_{w\in X^{\ast }},\times ,1),$$

$$\forall u\in Y^{\ast }-y_1{Y}^{\ast },\zeta (u):={\mathrm{H}}_u(+\infty )$$

$$\mathrm{L}:=\sum _{w\in X^{\ast }}{\mathrm{Li}}_{w}w=\prod _{l\in \mathcal{L}ynX}^{\searrow }{e}^{{\mathrm{Li}}_{S_l}{P}_l},$$

$$\displaystyle Z_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}:=\prod_{l\in{\mathcal{L}yn}Y-\{y_{1}\}}^{\searrow}e^{\mathrm{H}_{\Sigma_{l}}(+\infty)\Pi_{l}}$$

$$\displaystyle\zeta:{\displaystyle({\mathbb{Q}}1_{X^{*}}\oplus x_{0}{\mathbb{Q}}\langle X\rangle x_{1},\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})\atop\displaystyle({\mathbb{Q}}1_{Y^{*}}\oplus(Y-\{y_{1}\}){\mathbb{Q}}\langle Y\rangle,\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;,1_{Y^{*}})}$$

$$\genfrac{}{}{0ex}{}{x_0{x}_1^{r_1-1}\dots x_0{x}_1^{r_k-1}}{y_{s_1}\dots y_{s_k}}$$

$$\sum _{n\ge 0}{\mathrm{Li}}_{x_0^n}(z)t_0^n=z^{t_0}$$

$${\mathrm{Li}}_S(z)=\sum _{n\ge 0}⟨S\mid x_0^n⟩\frac{{\log }^n(z)}{n!}+\sum _{k\ge 1}\sum _{w\in (x_0^{\ast }{x}_1{)}^k{x}_0^{\ast }}⟨S\mid w⟩{\mathrm{Li}}_w(z)$$

$${\mathrm{Li}}_{(a{x}_0{)}^{\ast i}}(z)$$

$${\mathrm{Li}}_{(a{x}_1{)}^{\ast i}}(z)$$

$${\mathrm{Li}}_V(z)=\sum _{n_1>\dots >n_r>0}\frac{z^{n_1}}{(n_1-t_1{)}^{s_1}\dots (n_r-t_r{)}^{s_r}}.$$

$${\mathrm{Li}}_V(z)$$

$$=$$

$$\{{\mathrm{Li}}_S{\}}_{S\in \mathbb{C}⟨X⟩{}_{{}^{\bigsqcup \bigsqcup }}\mathbb{C}[x_0^{\ast }]{}_{{}^{\bigsqcup \bigsqcup }}\mathbb{C}[(-x_0^{\ast })]{}_{{}^{\bigsqcup \bigsqcup }}\mathbb{C}[x_1^{\ast }]}$$

$$\subset$$

$$\{{\mathrm{Li}}_S{\}}_{S\in \mathbb{C}⟨X⟩{}_{{}^{\bigsqcup \bigsqcup }}{\mathbb{C}}^{\mathrm{rat}}⟨⟨x_0⟩⟩{}_{{}^{\bigsqcup \bigsqcup }}{\mathbb{C}}^{\mathrm{rat}}⟨⟨x_1⟩⟩}$$

$$\subset$$

$$(D{E}_2)$$

$$(D{E}^{(1)})\mathbf{d}S=M_{1}S(D{E}^{(2)})\mathbf{d}S=S{M}_2.$$

$$\langle\mathrm{L}\mid w\rangle=\left\{\begin{array}[]{rcl}\displaystyle\frac{\log^{n}(z)}{n!}&\mbox{if}&w=x_{0}^{n}\cr\int_{0}^{z}\Big{(}(\frac{x_{1}}{1-z})\langle\mathrm{L}\mid u\rangle\Big{)}[s]\,ds&\mbox{if}&w=x_{1}u\cr\int_{0}^{z}\Big{(}(\frac{x_{0}}{z})\langle\mathrm{L}\mid ux_{1}x_{0}^{n}\rangle\Big{)}[s]\,ds&\mbox{if}&w=x_{0}ux_{1}x_{0}^{n}.\end{array}\right.$$

$$T^{\prime }=(\frac{x_0}{z}+\frac{x_1}{1-z})T+T(\frac{x_0}{z})$$

$$\displaystyle\displaystyle\lim_{z\rightarrow 1}\exp\biggl{(}-y_{1}\log\frac{1}{1-z}\biggr{)}\pi_{Y}(\mathrm{L}(z))$$

$$=$$

$$\displaystyle Z_{\gamma}=\exp(\gamma y_{1})\displaystyle\prod_{l\in{\mathcal{L}yn}Y-\{y_{1}\}}^{\searrow}\exp(\zeta(\Sigma_{l})\Pi_{l})=\exp(\gamma y_{1})Z_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}.$$

Full text

On Drinfel’d associators

G. H. E. Duchamp*♯, V. Hoang Ngoc Minh◆, K. A. Penson♭*

*♯*Université Paris XIII, 99 Jean-Baptiste Clément, 93430 Villetaneuse, France.

*◆*Université Lille II, 1, Place Déliot, 59024 Lille, France.

*♭*Université Paris VI, 75252 Paris Cedex 05, France.

1 Knizhnik-Zamolodchikov differential equations and coefficients of Drinfel’d associators

In 1986 [6], in order to study the linear representations of the braid group $B_{n}$ coming from the monodromy of the Knizhnik-Zamolodchikov differential equations, Drinfel’d introduced a class of formal power series $\Phi$ on noncommutative variables over the finite alphabet $X=\{x_{0},x_{1}\}$. Such a power series $\Phi$ is called an associator. For $n=3$, it leads to the following fuchsian noncommutative differential equation with three regular singularities in $\{0,1,+\infty\}$ :

[TABLE]

Solutions of $(DE)$ are power series, with coefficients which are mono-valued functions on the simply connex domain $\Omega={\mathbb{C}}-(]-\infty,0]\cup[1,+\infty[)$ and can be seen as multi-valued over111In fact, we have mappings from the connected covering $\widetilde{{\mathbb{C}}-\{0,1\}}$. ${{\mathbb{C}}-\{0,1\}}$, on noncommutative variables $x_{0}$ and $x_{1}$. Drinfel’d proved that $(DE)$ admits two particular mono-valued solutions on $\Omega$, $G_{0}(z)\;{}_{\widetilde{z\rightsquigarrow 0}}\;\exp[x_{0}\log(z)]$ and $G_{1}(z)\;{}_{\widetilde{z\rightsquigarrow 1}}\;\exp[-x_{1}\log(1-z)]$ [7, 8]. and the existence of an associator $\Phi_{KZ}\in{\mathbb{R}}\langle\!\langle X\rangle\!\rangle$ such that $G_{0}=G_{1}\Phi_{KZ}$ [7, 8]. After that, via representations of the chord diagram algebras, Lê and Murakami [17] expressed the coefficients of $\Phi_{KZ}$ as linear combinations of special values of several complex variables zeta functions, $\{\zeta_{r}\}_{r\in{\mathbb{N}}_{+}}$,

[TABLE]

where ${\mathcal{H}}_{r}=\{(s_{1},\ldots,s_{r})\in{\mathbb{C}}^{r}|\forall m=1,..,r,\sum_{i=1}^{m}\Re(s_{i})>m\}$. For $(s_{1},\ldots,s_{r})\in{\mathcal{H}}_{r}$, one has two ways of thinking $\zeta_{r}(s_{1},\ldots,s_{r})$ as limits, fulfilling identities [14, 13, 1]. Firstly, they are limits of polylogarithms and secondly, as truncated sums, they are limits of harmonic sums, for $z\in{\mathbb{C}},\mid\!z\!\mid<1,N\in{\mathbb{N}}_{+}$ :

[TABLE]

More precisely, if $(s_{1},\ldots,s_{r})\in{\mathcal{H}}_{r}$ then, after a theorem by Abel, one has

[TABLE]

else it does not hold, for $(s_{1},\ldots,s_{r})\notin{\mathcal{H}}_{r}$, while $\operatorname{Li}_{s_{1},\ldots,s_{k}}$ is well defined over $\{z\in{\mathbb{C}},\mid\!z\!\mid<1\}$ and so is $\mathrm{H}_{s_{1},\ldots,s_{k}}$, as Taylor coefficients of the following function

[TABLE]

Note also that, for $r=1$, $\zeta_{1}$ is nothing else but the famous Riemann zeta function and, for $r=0$, it is convenient to set $\zeta_{0}$ to the constant function $1_{{\mathbb{R}}}$. In all the sequel, for simplification, we will adopt the notation $\zeta$ for $\zeta_{r},r\in{\mathbb{N}}$.

In this work, we will describe the regularized solutions of $(DE)$.

For that, we are considering the alphabets $X=\{x_{0},x_{1}\}$ and $Y_{0}=\{y_{s}\}_{s\geq 0}$ equipped of the total ordering $x_{0}<x_{1}$ and $y_{0}>y_{1}>y_{2}>\ldots$, respectively. Let $Y=Y_{0}-\{y_{0}\}$. The free monoid generated by $X$ (resp. $Y,Y_{0}$) is denoted by $X^{*}$ (resp. $Y^{*},Y_{0}^{*}$) and admits $1_{X^{*}}$ (resp. $1_{Y^{*}},1_{Y_{0}^{*}}$) as unit.

The sets of, respectively, polynomials and formal power series, with coefficients in a commutative ${\mathbb{Q}}$-algebra $A$, over $X^{*}$ (resp. $Y^{*},Y_{0}^{*}$) are denoted by $A\langle X\rangle$ (resp. $A\langle Y\rangle,A\langle Y_{0}\rangle$) and $A\langle\!\langle X\rangle\!\rangle$ (resp. $A\langle\!\langle Y\rangle\!\rangle,A\langle\!\langle Y_{0}\rangle\!\rangle$). The sets of polynomials are the $A$-modules and endowed with the associative concatenation, the associative commutative shuffle (resp. quasi-shuffle) product, over $A\langle X\rangle$ (resp. $A\langle Y\rangle,A\langle Y_{0}\rangle$). Their associated coproducts are denoted, respectively, $\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$ and \Delta_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}. The algebras $(A\langle X\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})$ and (A\langle Y\rangle,\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;,1_{Y^{*}}) admit the sets of Lyndon words denoted, respectively, by ${\mathcal{L}yn}X$ and ${\mathcal{L}yn}Y$, as transcendence bases [18] (resp. [15, 16]).

For $Z=X$ or $Y$, denoting ${\cal L}ie_{A}\langle Z\rangle$ and ${\cal L}ie_{A}\langle\!\langle Z\rangle\!\rangle$ the sets of, respectively, Lie polynomials and Lie series, the enveloping algebra $\mathcal{U}({\cal L}ie_{A}\langle Z\rangle)$ is isomorphic to the Hopf algebra $(A\langle X\rangle,.,1_{Z^{*}},\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}},{\tt e})$. We get also {\mathcal{H}}_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}:=(A\langle Y\rangle,.,1_{Y^{*}},\Delta_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;},{\tt e})\cong\mathcal{U}(\mathrm{Prim}({\mathcal{H}}_{{\scriptstyle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}})), where [15, 16]

[TABLE]

2 Indexing polylogarithms and harmonic sums by words and their generating series

For any $r\in{\mathbb{N}}$, since any combinatorial composition $(s_{1},\ldots,s_{r})\in{\mathbb{N}}_{+}^{r}$ can be associated with words $x_{0}^{s_{1}-1}x_{1}\ldots x_{0}^{s_{r}-1}x_{1}\in X^{*}x_{1}$ and $y_{s_{1}}\ldots y_{s_{r}}\in Y^{*}$. Similarly, any multi-indice222The weight of $(s_{1},\ldots,s_{r})\in{\mathbb{N}}_{+}^{r}$ (resp. ${\mathbb{N}}^{r}$) is defined as the integer $s_{1}+\ldots+s_{r}$ which corresponds to the weight, denoted $(w)$, of its associated word $w\in Y^{*}$ (resp. $Y_{0}^{*}$) and corresponds also to the length, denoted by $|u|$, of its associated word $u\in X^{*}$. $(s_{1},\ldots,s_{r})\in{\mathbb{N}}^{r}$ can be associated with words $y_{s_{1}}\ldots y_{s_{r}}\in Y_{0}^{*}$. Then let $\operatorname{Li}_{x_{0}^{r}}(z):={(\log(z))^{r}}/{r!}$, and $\operatorname{Li}_{s_{1},\ldots,s_{k}}$ and $\mathrm{H}_{s_{1},\ldots,s_{k}}$ be indexed by words [14] : $\operatorname{Li}_{x_{0}^{s_{1}-1}x_{1}\ldots x_{0}^{s_{r}-1}x_{1}}:=\operatorname{Li}_{s_{1},\ldots,s_{r}}$ and $\mathrm{H}_{y_{s_{1}}\ldots y_{s_{r}}}:=\mathrm{H}_{s_{1},\ldots,s_{r}}$. Similarly, $\operatorname{Li}_{-s_{1},\ldots,-s_{k}}$ and $\mathrm{H}_{-s_{1},\ldots,-s_{k}}$ be indexed by words333Note that, all these $\{\operatorname{Li}^{-}_{w}\}_{w\in Y_{0}^{*}}$ and $\{\mathrm{H}^{-}_{w}\}_{w\in Y_{0}^{*}}$ are divergent at their singularities. [4, 5] : $\operatorname{Li}^{-}_{y_{s_{1}}\ldots y_{s_{r}}}:=\operatorname{Li}_{-s_{1},\ldots,-s_{r}}$ and $\mathrm{H}^{-}_{y_{s_{1}}\ldots y_{s_{r}}}:=\mathrm{H}_{-s_{1},\ldots,-s_{r}}$. In particular, $\mathrm{H}^{-}_{y_{0}^{r}}(n):={n\choose r}=(n)_{r}/r!$ and $\operatorname{Li}^{-}_{y_{0}^{r}}(z):=({z}/(1-z))^{r}$. There exists a law of algebra, denoted by $\top$, in ${\mathbb{Q}}\langle\!\langle Y_{0}\rangle\!\rangle$, such that he following morphisms of algebras are surjective [4]

[TABLE]

and $\ker\mathrm{H}^{-}_{\bullet}=\ker\operatorname{Li}^{-}_{\bullet}={\mathbb{Q}}\langle\{w-w\top 1_{Y_{0}^{*}}|w\in Y_{0}^{*}\}\rangle$ [4]. Moreover, the families $\{\mathrm{H}^{-}_{y_{k}}\}_{k\geq 0}$ and $\{\operatorname{Li}^{-}_{y_{k}}\}_{k\geq 0}$ are ${\mathbb{Q}}$-linearly independent.

On the other hand, the following morphisms of algebras are injective

[TABLE]

Moreover, the families $\{\mathrm{H}_{w}\}_{w\in Y^{*}}$ and $\{\operatorname{Li}_{w}\}_{w\in X^{*}}$ are ${\mathbb{Q}}$-linearly independent and the families $\{\mathrm{H}_{l}\}_{l\in{\mathcal{L}yn}Y}$ and $\{\operatorname{Li}_{l}\}_{l\in{\mathcal{L}yn}X}$are ${\mathbb{Q}}$-algebraically independent. But at singularities of $\{\operatorname{Li}_{w}\}_{w\in X^{*}},\{\mathrm{H}_{w}\}_{w\in Y^{*}}$, the following convergent values

[TABLE]

are no longer linearly independent and the values $\{\mathrm{H}_{l}(+\infty)\}_{l\in{\mathcal{L}yn}Y-\{y_{1}\}}$ (resp. $\{\operatorname{Li}_{l}(1)\}_{l\in{\mathcal{L}yn}X-X}$) are no longer algebraically independent [14, 19].

The graphs of the isomorphisms of algebras, $\operatorname{Li}_{\bullet}$ and $\mathrm{H}_{\bullet}$, as generating series, read then [2, 14]

[TABLE]

where the PBW basis $\{P_{w}\}_{w\in X^{*}}$ (resp. $\{\Pi_{w}\}_{w\in Y^{*}}$) is expanded over the basis of ${\cal L}ie_{A}\langle X\rangle$ (resp. \mathcal{U}(\mathrm{Prim}({\mathcal{H}}_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;})), $\{P_{l}\}_{l\in{\mathcal{L}yn}X}$ (resp. $\{\Pi_{l}\}_{l\in{\mathcal{L}yn}Y}$), and $\{S_{w}\}_{w\in X^{*}}$ (resp. $\{\Sigma_{w}\}_{w\in Y^{*}}$) is the basis of $({\mathbb{Q}}\langle Y\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})$ (resp. ({\mathbb{Q}}\langle Y\rangle,\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;,1_{Y^{*}})) containing the transcendence basis $\{S_{l}\}_{l\in{\mathcal{L}yn}X}$ (resp. $\{\Sigma_{l}\}_{l\in{\mathcal{L}yn}Y}$).

By termwise differentiation, $\mathrm{L}$ satisfies the noncommutative differential equation $(DE)$ with the boundary condition $\mathrm{L}(z){}_{\widetilde{z\to 0^{+}}}e^{x_{0}\log(z)}$. It is immediate that the power series $\mathrm{H}$ and $\mathrm{L}$ are group-like, for \Delta_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;} and $\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$, respectively. Hence, the following noncommutative generating series are well defined and are group-like, for \Delta_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;} and $\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$, respectively [14, 15, 16] :

[TABLE]

Definitions (3) and (11) lead then to the following surjective poly-morphism

[TABLE]

where ${\mathcal{Z}}$ is the ${\mathbb{Q}}$-algebra generated by $\{\zeta(l)\}_{l\in{\mathcal{L}yn}X-X}$ (resp. $\{\zeta(S_{l})\}_{l\in{\mathcal{L}yn}X-X}$), or equivalently, generated by $\{\zeta(l)\}_{l\in{\mathcal{L}yn}Y-\{y_{1}\}}$ (resp. $\{\zeta(\Sigma_{l})\}_{l\in{\mathcal{L}yn}Y-\{y_{1}\}}$).

Now, let $t_{i}\in{\mathbb{C}},\mid\!t_{i}\!\mid<1,i\in{\mathbb{N}}$. For $z\in{\mathbb{C}},\mid\!z\!\mid<1$, we have [11]

[TABLE]

These suggest to extend the morphism $\operatorname{Li}_{\bullet}$ over $(\mathrm{Dom}(\operatorname{Li}_{\bullet}),\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})$, via Lazard’s elimination, as follows (subjected to be convergent)

[TABLE]

with ${\mathbb{C}}\langle X\rangle\mathop{{}_{{}^{\sqcup\!\sqcup}}}{\mathbb{C}}^{\mathrm{rat}}\langle\!\langle x_{0}\rangle\!\rangle\mathop{{}_{{}^{\sqcup\!\sqcup}}}{\mathbb{C}}^{\mathrm{rat}}\langle\!\langle x_{1}\rangle\!\rangle\subset\mathrm{Dom}(\operatorname{Li}_{\bullet})\subset{\mathbb{C}}^{\mathrm{rat}}\langle\!\langle X\rangle\!\rangle$ and ${\mathbb{C}}^{\mathrm{rat}}\langle\!\langle X\rangle\!\rangle$ denotes the closure, of ${{\mathbb{C}}}\langle X\rangle$ in ${\mathbb{C}}\langle\!\langle X\rangle\!\rangle$, by $\{+,.,{}^{*}\}$. For example [11, 12],

For any $i,j\in{\mathbb{N}}_{+}$ and $x\in X$, since $(t_{0}x_{0}+t_{1}x_{1})^{*}=(t_{0}x_{0})^{*}\mathop{{}_{{}^{\sqcup\!\sqcup}}}(t_{1}x_{1})^{*}$ and $(x^{*})^{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}i}=(ix)^{*}$ then $\operatorname{Li}_{(x_{0}^{*})^{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}i}\mathop{{}_{{}^{\sqcup\!\sqcup}}}(x_{1}^{*})^{{\scriptstyle\mathop{{}_{{}^{\sqcup\!\sqcup}}}}j}}(z)=z^{i}(1-z)^{-j}$. 2. 2.

For $a\in{\mathbb{C}},x\in X,i\in{\mathbb{N}}_{+}$, since $(ax)^{*i}=(ax)^{*}\mathop{{}_{{}^{\sqcup\!\sqcup}}}(1+ax)^{i-1}$ then

[TABLE] 3. 3.

Let $V=(t_{1}x_{0})^{*s_{1}}x_{0}^{s_{1}-1}x_{1}\ldots(t_{r}x_{0})^{*s_{r}}x_{0}^{s_{r}-1}x_{1}$, for $(s_{1},\ldots,s_{r})\in{\mathbb{N}}_{+}^{r}$. Then

[TABLE]

In particular, for $s_{1}=\ldots=s_{r}=1$, then one has

[TABLE] 4. 4.

From the previous points, one has

[TABLE]

3 Noncommutative evolution equations

As we said previously Drinfel’d proved that $(DE)$ admits two particular solutions on $\Omega$. These new tools and results can be considered as pertaining to the domain of noncommutative evolution equations. We will, here, only mention what is relevant for our needs.

Even for one sided 444As the left (DE) for instance. differential equations, in order to cope with limit initial conditions (see applications below), one needs the two sided version.

Let then $\Omega\subset{\mathbb{C}}$ be simply connected and open and ${\mathcal{H}}(\Omega)$ denote the algebra of holomorphic functions on $\Omega$. We suppose given two series (called multipliers) $M_{1},M_{2}\in{\mathcal{H}}(\Omega)_{+}\langle\!\langle X\rangle\!\rangle$ ($X$ is an alphabet and the subscript indicates that the series have no constant term). Let then

[TABLE]

be our equation.

3.1 The main theorem

Theorem 1**.**

Let

[TABLE]

1. (i)

Solutions of $(DE_{2})$ form a ${\mathbb{C}}$-vector space. 2. (ii)

Solutions of $(DE_{2})$ have their constant term (as coefficient of $1_{X^{*}}$) which are constant functions (on $\Omega$); there exists solutions with constant coefficient $1_{\Omega}$ (hence invertible). 3. (iii)

If two solutions coincide at one point $z_{0}\in\overline{\Omega}$, they coincide everywhere. 4. (iv)

Let be the following one-sided equations

[TABLE]

and let $S_{i},\ i=1,2$ a solution of $(DE^{(i)})$, then $S_{1}S_{2}$ is a solution of $(DE_{2})$. Conversely, every solution of $(DE_{2})$ can be constructed so. 5. (v)

If $M_{i},\ i=1,2$ are primitive and if $S$, a solution of $(DE_{2})$, is group-like at one point, (or, even at one limit point) it is globally group-like.

Proof.

Omitted. ∎

Remark 1**.**

• •

Every holomorphic series $S(z)\in{\mathcal{H}}(\Omega)\langle\!\langle X\rangle\!\rangle$ which is group-like ($\Delta(S)=S\otimes S$* and $\langle S\mid 1_{X^{*}}\rangle$) is a solution of a left-sided dynamics with primitive multiplier (take $M_{1}=\mathbf{d}(S)S^{-1}$ and $M_{2}=0$).*

• •

Invertible solutions of an equation of type $S^{\prime}=M_{1}S$ are on the same orbit by multiplication on the right by invertible constant series i.e. let $S_{i},\ i=1,2$ be invertible solutions of $(DE^{(1)})$, then there exists an unique invertible $T\in{\mathbb{C}}\langle\!\langle X\rangle\!\rangle$ such that $S_{2}=S_{1}.T$. From this and point (iv) of the theorem, one can parametrize the set of invertible solutions of $(DE_{2})$.

3.2 Application: Unicity of solutions with asymptotic conditions.

In a previous work [3], we proved that asymptotic group-likeness, for a series, implies555Under the condition that the multiplier be primitive, result extended as point (v) of the theorem above. that the series in question is group-like everywhere. The process above (theorem (1), Picard’s process) can be performed, under certain conditions with improper integrals we then construct the series $\mathrm{L}$ recursively as

[TABLE]

one can check that

• •

this process is well defined at each step and computes the series $\mathrm{L}$ as below.

• •

$\mathrm{L}$ is solution of $(DE)$, is exactly $G_{0}$ and is group-like

We here only prove that $G_{0}$ is unique using the theorem above. Consider the series $T=Le^{-x_{0}\log(z)}$. Then $T$ is solution of an equation of the type $(DE_{2})$

[TABLE]

but $\lim_{z\to z_{0}}G_{0}e^{-x_{0}\log(z)}=1$ so, by the point (iii) of theorem (1) one has $G_{0}e^{-x_{0}\log(z)}=Le^{-x_{0}\log(z)}$ and then $G_{0}=L$.

A similar (and symmetric) argument can be performed for $G_{1}$ and then, in this interpretation and context, $\Phi_{KZ}$ is unique.

4 Double global regularization of associators

Global singularities analysis leads to to the following global renormalization [2]

[TABLE]

Thus, the coefficients $\{\langle Z_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}|u\rangle\}_{u\in X^{*}}$ (i.e. $\{\zeta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}(u)\}_{u\in X^{*}}$) and \{\langle Z_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}|v\rangle\}_{v\in Y^{*}} (i.e. \{\zeta_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}(v)\}_{v\in Y^{*}}) represent the finite part of the asymptotic expansions, in $\{(1-z)^{-a}\log^{b}(1-z)\}_{a,b\in{\mathbb{N}}}$ (resp. $\{n^{-a}\mathrm{H}_{1}^{b}(n)\}_{a,b\in{\mathbb{N}}}$) of $\{\operatorname{Li}_{w}\}_{u\in X^{*}}$ (resp. $\{\mathrm{H}_{w}\}_{v\in Y^{*}}$). On the other way, by a transfer theorem [10], let $\{\gamma_{w}\}_{v\in Y^{*}}$ be the finite parts of $\{\mathrm{H}_{w}\}_{v\in Y^{*}}$, in $\{n^{-a}\log^{b}(n)\}_{a,b\in{\mathbb{N}}}$, and let $Z_{\gamma}$ be their noncommutative generating series. The map \gamma_{\bullet}:({\mathbb{Q}}\langle Y\rangle,\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;,1_{Y^{*}})\rightarrow({\cal Z},\times,1), mapping $w$ to $\gamma_{w}$, is then a character and $Z_{\gamma}$ is group-like, for \Delta_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}. Moreover [15, 16],

[TABLE]

The asymptotic behavior leads to the bridge666This equation is different from Jean Écalle’s one [9]. equation [2, 15, 16]

[TABLE]

where $B(y_{1})=\exp(\gamma y_{1}-\sum_{k\geq 2}(-y_{1})^{k}{\zeta(k)}/k)$ and $B^{\prime}(y_{1})=\exp(-\gamma y_{1})B(y_{1})$.

Similarly, there is $C^{-}_{w}\in{\mathbb{Q}}$ and $B^{-}_{w}\in{\mathbb{N}}$, such that $\mathrm{H}^{-}_{w}(N){}_{\widetilde{N\rightarrow+\infty}}N^{(w)+\mid\!w\!\mid}C^{-}_{w}$ and $\operatorname{Li}^{-}_{w}(z){}_{\widetilde{z\rightarrow 1}}(1-z)^{-(w)-\mid\!w\!\mid}B^{-}_{w}$ [4]. Moreover,

[TABLE]

Now, one can then consider the following noncommutative generating series :

[TABLE]

Then $\mathrm{H}^{-}$ and $C^{-}$ are group-like for, respectively, \Delta_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;} and $\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$ and [4]

[TABLE]

Next, for any $w\in Y_{0}^{*}$, there exists then a unique polynomial $p\in({\mathbb{Z}}[t],\times,1)$ of degree $(w)+\mid\!w\!\mid$ such that [4]

[TABLE]

where777Here, it is also convenient to denote ${\mathbb{Q}}[(n)_{\bullet}]$ the set of “polynomials” expanded as follows

$\displaystyle\forall p\in,{\mathbb{Q}}[(n)_{\bullet}],$ $\displaystyle p=\displaystyle\sum_{k=0}^{d}p_{k}(n)_{k},$ \displaystyle\mathop{\mathrm{missing}}{deg}\nolimits(p)=d.

where $(n)_{\bullet}:{\mathbb{N}}\longrightarrow{\mathbb{Q}}$ mapping $i$ to $(n)_{i}=n(n-1)\ldots(n-i+1)$. In other terms, for any $w\in Y_{0}^{*},k\in{\mathbb{N}},0\leq k\leq(w)+\mid\!w\!\mid$, one has ${\langle\operatorname{Li}^{-}_{w}\mid(1-z)^{-k}\rangle}={k!}\langle\mathrm{H}^{-}_{w}\mid(n)_{k}\rangle$.

Hence, denoting $\tilde{p}$ the exponential transformed of the polynomial $p$, one has $\operatorname{Li}^{-}_{w}(z)=p((1-z)^{-1})$ and $\mathrm{H}^{-}_{w}(n)=\tilde{p}((n)_{\bullet})$ with

[TABLE]

Let us then associate $p$ and $\tilde{p}$ with the polynomial $\check{p}$ obtained as follows

[TABLE]

Let us recall also that, for any $c\in{\mathbb{C}}$, one has $(n)_{c}{}_{\widetilde{n\rightarrow+\infty}}n^{c}=e^{c\log(n)}$ and, with the respective scales of comparison, one has the following finite parts

[TABLE]

Hence, using the notations given in (40) and (41), one can see, from (44) and (45), that the values $p(1)$ and $\tilde{p}(1)$ obtained in (42) represent

[TABLE]

One can use then these values $p(1)$ and $\tilde{p}(1)$, instead of the values $B^{-}_{w}$ and $C^{-}_{w}$, to regularize, respectively, $\zeta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}(R_{w})$ and $\zeta_{\gamma}(\pi_{Y}(R_{w}))$ as showed Theorem 2 bellow because, essentially, $B^{-}_{\bullet}$ and $C^{-}_{\bullet}$ do not realize characters for, respectively, $({\mathbb{Q}}\langle X\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}},\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}},{\tt e})$ and ({\mathbb{Q}}\langle Y\rangle,\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;,1_{Y^{*}},\Delta_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;},{\tt e}) [4].

Now, in virtue of the extension of $\operatorname{Li}_{\bullet}$, defined as in (16) and (17), and of the Taylor coefficients, the previous polynomials $p,\tilde{p}$ and $\check{p}$ given in (42)–(43) can be determined explicitly thanks to

Proposition 1**.**

The following morphisms of algebras are bijective

[TABLE] 2. 2.

For any $w=y_{s_{1}},\ldots y_{s_{r}}\in Y_{0}^{*}$, there exists a unique polynomial $R_{w}$ belonging to $({\mathbb{Z}}[x_{1}^{*}],\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})$ of degree $(w)+\mid\!w\!\mid$, such that

[TABLE]

In particular, via the extension, by linearity, of $R_{\bullet}$ over ${\mathbb{Q}}\langle Y_{0}\rangle$ and via the linear independent family $\{\operatorname{Li}^{-}_{y_{k}}\}_{k\geq 0}$ in ${\mathbb{Q}}\{\operatorname{Li}^{-}_{w}\}_{w\in Y_{0}^{*}}$, one has

[TABLE] 3. 3.

For any $w$, one has $\check{p}(x_{1}^{*})=R_{w}$. 4. 4.

More explicitly, for any $w=y_{s_{1}},\ldots y_{s_{r}}\in Y_{0}^{*}$, there exists a unique polynomial $R_{w}$ belonging to $({\mathbb{Z}}[x_{1}^{*}],\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})$ of degree $(w)+\mid\!w\!\mid$, given by

[TABLE]

where, for any $i=1,\ldots,r$, if ${k_{i}}=0$ then $\rho_{k_{i}}=x_{1}^{*}-1_{X^{*}}$ else, for ${k_{i}}>0$, denoting the Stirling numbers of second kind by $S_{2}(k,j)$’s, one has

[TABLE]

Proposition 2** ([2, 15, 16]).**

With notations of (14), similar to the character $\gamma_{\bullet}$, the poly-morphism $\zeta$ can be extended as follows

[TABLE]

satisfying, for any $l\in{\mathcal{L}yn}Y-\{y_{1}\}$, \zeta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}(\pi_{X}(l))=\zeta_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}(l)=\gamma_{l}=\zeta(l) and, for the generators of length (resp. weight) one, for $X^{*}$ (resp. $Y^{*}$), $\gamma_{y_{1}}=\gamma$ and \zeta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}(x_{0})=\zeta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}(x_{1})=\zeta_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;}(y_{1})=0.

Now, to regularize $\{\zeta(s_{1},\ldots,s_{r})\}_{(s_{1},\ldots,s_{r})\in{\mathbb{C}}^{r}}$, we use

Lemma 1** ([4]).**

The power series $x_{0}^{*}$ and $x_{1}^{*}$ are transcendent over ${\mathbb{C}}\langle X\rangle$. 2. 2.

The family $\{x_{0}^{*},x_{1}^{*}\}$ is algebraically independent over $({\mathbb{C}}\langle X\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})$ within $({\mathbb{C}}\langle\!\langle X\rangle\!\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})$. 3. 3.

The module $({\mathbb{C}}\langle X\rangle,\mathop{{}_{{}^{\sqcup\!\sqcup}}},1_{X^{*}})[x_{0}^{*},x_{1}^{*},(-x_{0})^{*}]$ is ${\mathbb{C}}\langle X\rangle$-free and the family $\{(x_{0}^{*})^{\mathop{{}_{{}^{\sqcup\!\sqcup}}}k}\mathop{{}_{{}^{\sqcup\!\sqcup}}}(x_{1}^{*})^{\mathop{{}_{{}^{\sqcup\!\sqcup}}}l}\}^{(k,l)\in{\mathbb{Z}}\times{\mathbb{N}}}$ forms a ${\mathbb{C}}\langle X\rangle$-basis of it.

Hence, $\{w\mathop{{}_{{}^{\sqcup\!\sqcup}}}(x_{0}^{*})^{\mathop{{}_{{}^{\sqcup\!\sqcup}}}k}\mathop{{}_{{}^{\sqcup\!\sqcup}}}(x_{1}^{*})^{\mathop{{}_{{}^{\sqcup\!\sqcup}}}l}\}^{(k,l)\in{\mathbb{Z}}\times{\mathbb{N}}}_{w\in X^{*}}$ is a ${\mathbb{C}}$-basis of it. 4. 4.

One has, for any $x_{i}\in X$, ${\mathbb{C}}^{\mathrm{rat}}\langle\!\langle x_{i}\rangle\!\rangle=\mathrm{span}_{{\mathbb{C}}}\{(tx_{i})^{*}\mathop{{}_{{}^{\sqcup\!\sqcup}}}{{\mathbb{C}}}\langle x_{i}\rangle|{t\in{\mathbb{C}}}\}$.

Since, for any $t\in{\mathbb{C}},\mid\!t\!\mid<1$, one has $\operatorname{Li}_{(tx_{1})^{*}}(z)=(1-z)^{-t}$ and

[TABLE]

then, with the notations of Proposition 2, we extend extend the characters $\zeta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$ and $\gamma_{\bullet}$, defined in Proposition 2, over ${\mathbb{C}}\langle X\rangle\mathop{{}_{{}^{\sqcup\!\sqcup}}}{\mathbb{C}}[x_{1}^{*}]$ and {\mathbb{C}}\langle Y\rangle\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;{\mathbb{C}}[y_{1}^{*}], respectively, as follows

Proposition 3** ([4]).**

The characters $\zeta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$ and $\gamma_{\bullet}$ can be extended as follows

[TABLE]

Therefore, in virtue of Propositions 1 and 3, we obtain then

Theorem 2**.**

For any $(s_{1},\ldots,s_{r})\in{\mathbb{N}}_{+}^{r}$ associated with $w\in Y^{*}$, there exists a unique polynomial $p\in{\mathbb{Z}}[t]$ of valuation $1$ and of degree $(w)+\mid\!w\!\mid$ such that

[TABLE] 2. 2.

Let $\Upsilon(n)\in{\mathbb{Q}}[(n)_{\bullet}]\langle\!\langle Y\rangle\!\rangle$ and $\Lambda(z)\in{\mathbb{Q}}[(1-z)^{-1}][\log(z)]\langle\!\langle X\rangle\!\rangle$ be the noncommutative generating series of $\{\mathrm{H}_{\pi_{Y}(R_{w})}\}_{w\in Y^{*}}$ and $\{\operatorname{Li}_{R_{\pi_{Y}(w)}}\}_{w\in X^{*}}$ :

[TABLE]

Then $\Upsilon$ and $\Lambda$ are group-like, for respectively \Delta_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;} and $\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$, and :

[TABLE] 3. 3.

Let $Z^{-}_{\gamma}\in{\mathbb{Q}}\langle\!\langle Y\rangle\!\rangle$ and $Z^{-}_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}\in{\mathbb{Z}}\langle\!\langle X\rangle\!\rangle$ be the noncommutative generating series of $\{\gamma_{\pi_{Y}(R_{w})}\}_{w\in Y^{*}}$ and888On the one hand, by Proposition 2, one has $\langle Z^{-}_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}\mid x_{0}\rangle=\zeta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}(x_{0})=0$. On the other hand, since $R_{y_{1}}=(2x_{1})^{*}-x_{1}^{*}$ then $\operatorname{Li}_{R_{y_{1}}}(z)=(1-z)^{-2}-(1-z)^{-1}$ and $\mathrm{H}_{\pi_{Y}(R_{y_{1}})}(n)={n\choose 2}-{n\choose 1}$. Hence, one also has $\langle Z^{-}_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}\mid x_{1}\rangle=\zeta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}(R_{\pi_{Y}(y_{1})})=0$ and $\langle Z^{-}_{\gamma}\mid x_{1}\rangle=\gamma_{\pi_{Y}(R_{y_{1}})}=-1/2$.* $\{\zeta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}(R_{\pi_{Y}(w)})\}_{w\in X^{*}}$, respectively :*

[TABLE]

Then $Z^{-}_{\gamma}$ and $Z^{-}_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$ are group-like, for respectively \Delta_{\;\begin{picture}(20.0,10.0)(220.0,580.0)\put(220.0,592.0){\line(0,-1){10.0}} \put(220.0,582.0){\line(1,0){20.0}} \put(240.0,582.0){\line(0,1){10.0}} \put(230.0,592.0){\line(0,-1){10.0}} \put(225.0,587.0){\line(1,0){10.0}} \end{picture}\;} and $\Delta_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$, and :

[TABLE]

Moreover, $\mathrm{F.P.}_{n\rightarrow+\infty}\Upsilon(n)=Z^{-}_{\gamma}$ and $\mathrm{F.P.}_{z\rightarrow 1}\Lambda(z)=Z^{-}_{\mathop{{}_{{}^{\sqcup\!\sqcup}}}}$ meaning that, for any $v\in Y^{*}$ and $u\in X^{*}$, one has

[TABLE]