2002/02/15 by Georges Racinet, Racinet, Georges
Arts and Humanities · Mathematics · Social Sciences · #Algebraic Geometry (math.AG) #FOS: Mathematics #French Language Learning Methods #Historical Linguistics and Language Studies #Linguistics and Discourse Analysis #Quantum Algebra (math.QA) #math.AG #math.QA
paper · pdf · doi:10.48550/arxiv.math/0202142
41 pages, french, w/iso-latin-1 encoding. Uses mathrsfs and (included) French Math. Soc. document class smfart
arxiv created 2002/02/15 · openalex publication_date 2002/02/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The values at positive integers of the polyzeta functions are solutions of the polynomial equations arising from Drinfeld's associators, which have numerous applications in quantum algebra. Considered as iterated integrals they become periods of the motivic fundamental groupoid of ℙ1∖\0,1,∞\. From there comes a fundamental, yet no more explicit, system of algebraic relations; it implies the system of associators. We focus here on the combinatorics properties of another system of relations, the ``double shuffles'', which comes from elementary series and integrals manipulations. We show that it shares a torsor property with associators and ``motivic'' relations, is implied by the latter and defines a polynomial algebra over ℚ (a result first due to J. Ecalle). We obtain these results for more general numbers: values of Goncharov's multiple polylogarithms at roots of unity. These results were previously announced in math.QA:0012024. Here is the detailed proof.