Galois symmetries of fundamental groupoids and noncommutative geometry
type: Item
instance of: scholarly article
title: Galois symmetries of fundamental groupoids and noncommutative geometry
zbMATH Open document ID: 1095.11036
DOI: 10.1215/S0012-7094-04-12822-2
published in: Duke Mathematical Journal
publication date: 2005-09-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0208144
review text: The author introduces a Hopf algebra of motivic iterated integrals on the line, and proves an explicit formula for the coproduct \(\Delta\) which encodes the group law of the automorphism group of a certain noncommutative variety. He also relates the coproduct \(\Delta\) to the coproduct in the Hopf algebra of decorated rooted plane trivalent trees and give, as an application, explicit formulas for the coproduct in the motivic multiple polylogarithms Hopf algebra. These formulas play a key role in the striking correspondence between the motivic fundamental group of \(\mathbb{P}^1-(\{0,\infty\}^u\mu_N)\) (\(\mu_N\) being the group of all \(N\)th roots of unit) and modular varieties of \(\text{GL}_n\), which was previously investigated by the author. Finally, the author discusses some general principles relating Feynman integrals and mixed motives.
reviewed by: Masanori Morishita
Mathematics Subject Classification ID: 11G55
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 19E15
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 20F34
zbMATH DE Number: 2201102
zbMATH Keywords: Hopf algebra of motivic integrals
zbMATH Keywords: motivic multiple polylogarithm Hopf algebra
zbMATH Keywords: motivic fundamental groupoids
zbMATH Keywords: noncommutative geometry
zbMATH Keywords: Feynman integrals
author: A. B. Goncharov
MaRDI profile type: Publication
arXiv ID: math/0208144
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OpenAlex ID: W2086007622
description: scientific article