distribution relations {#sec:Rac}
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Multiple polylogs satisfy the following distribution formula (cf. [@Rac Prop. 2.25]): $$\label{equ:dist}
Li_{s_1,\dots,s_n}(x_1,\dots,x_n) = d^{s_1+\dots+s_n-n}
\sum_{y_j^d=x_j,1\le j\le n} Li_{s_1,\dots,s_n}(y_1,\dots,y_n),$$ for all positive integer $d$. When $s_1=1$ we need to exclude the case of $x_1=1$. We call these *finite distribution relations* (FDT). Racinet further considers the regularized version of these relations, which we now recall briefly.

Fix an embedding $\mmu_N\hookrightarrow \C$ and denote by $\gG$ its image. Define two sets of words $$\bfX:=\bfX_\gG=\{x_\gs: \gs\in \gG\cup\{0\}\}, \quad\text{and}\quad
 \bfY:=\bfY_\gG=\{y_{n,\gs}=x_0^{n-1}x_\gs: n\in \N, \gs\in \gG\}.$$ Then one may consider the coproduct $\gD$ of $\Q\langle\bfX\rangle$ defined by $\gD x_\gs=1\otimes x_\gs+x_\gs\otimes 1$ for all $\gs\in \gG\cup\{0\}$. For every path $\gam\in {{\mathbb P}}^1(\C)-(\{0,\infty\}\cup\gG)$ Racinet defines the group-like element $\calI_\gam\in \CXX$ by $$\calI_\gam:=\sum_{p\in \N,\gs_1,\dots,\gs_p\in  \gG\cup\{0\}}
 \calI_\gam(\gs_1,\dots,\gs_p)x_{\gs_1}\cdots x_{\gs_p},$$ where $\calI_\gam(\gs_1,\dots,\gs_p)$ is the iterated integral $\int_\gam \om(\gs_1) \cdots \om(\gs_p)$ with $$\om(\gs)(t)=\left\{
              \begin{array}{ll}
                \gs\,dt/(1-\gs t), & \hbox{if $\gs\ne 0$;} \\
                dt/t, & \hbox{if $\gs=0$.}
              \end{array}
            \right.$$ This $I_\gam$ is essentially the same element denoted by $\operatorname{{dch}}$ in [@DG]. Note that $\QY$ is the sub-algebra of $\QX$ generated by words not ending with $x_0$. We let $\pi_\bfY:\QX\ra \QY$ be the projection. As $x_0$ is primitive one knows that $(\QY,\gD)$ has a graded co-algebra structure.

Let $\QX_\cv$ be the sub-algebra 