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Double shuffles of multiple polylogarithms at roots of unity. (Doubles mélanges des polylogarithmes multiples aux racines de l’unité.) (French) Zbl 1050.11066
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 A. B. Goncharov [Duke Math J. 110, No.3, 397-487 (2991; Zbl 1113.14020)]

MSC:
11G55 Polylogarithms and relations with \(K\)-theory
11M41 Other Dirichlet series and zeta functions
33D70 Other basic hypergeometric functions and integrals in several variables
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