Double shuffles of multiple polylogarithms at roots of unity
The paper starts with the values at the positive integers of the multiple zeta functions \[ \zeta(s_1,\ldots,s_r)=\sum_{n_1>\cdots>n_r>0}{1\over n_1^{s_1}\cdots n_r^{s_r}}, \] the multiple polylogarithms \[ L_{s_1\ldots s_r}(z_1,\ldots,z_r)=\sum_{n_1>0}\sum_{n_1>n_2>\cdots>n_r} {z_1^{n_1}z_2^{n_2}\cdots z_r^{n_r}\over n_1^{s_1}n_2^{s_2}\cdots n_r^{s_r}}, \] and the iterated integrals \[ I_{[0,1]}(a_1,\ldots,a_p)=\int_{0\leq t_p\leq\cdots\leq t_1\leq1} \wedge_{i=1}^p \omega_{a_i}(t_i) \] with \(\omega_a(t)={dt\over a^{-1}-t}\) for \(a\neq 0\) and \(\omega_0={dt\over t}\). There are elementary combinatorial relations between these entities and the ultimate aim is to describe all such relations. One set of relations goes by the name of the `double shuffle'. An example of it arises from manipulation of the series to get the following equation \[ L_{s_1}(z_1)L_{s_2}(z_2)=L_{s_1s_2}(z_1,z_2)+L_{s_2s_1}(z_2,z_1) +L_{s_1+s_2}(z_1z_2) \] and another by decomposing the product of two simplices to get \[ I_{[0,1]}(a_1,\ldots,a_p)I_{[0,1]}(b_1,\ldots,b_q)= \sum_{\sigma\;\text{ in}\;{\mathcal S}_{p,q}} I_{[0,1]}(a_{\sigma^{-1}(1)},\ldots,a_{\sigma^{-1}(i)}) \] where \({\mathcal S}_{p,q}\) denotes the permutation of \(\{1,\ldots,p+q\}\) which increases on \(\{1,\ldots,p\}\) and on \(\{p+1,\ldots,p+q\}\). An additional relation called regularisation represents the resolution of certain singularities. The author sets out to put all this is a general algebraic setting related. Another account which includes concrete details of the constructions and a survey of other recent advances has been given by \textit{A. B. Goncharov} [Duke Math J. 110, No.3, 397-487 (2991; Zbl 1113.14020)]
scientific article; zbMATH DE number 1874455