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VOL. 63 | 2012 Combinatorics of the double shuffle Lie algebra
Sarah Carr, Leila Schneps

Editor(s) Hiroaki Nakamura, Florian Pop, Leila Schneps, Akio Tamagawa


In this article we give two combinatorial properties of elements satisfying the stuffle relations; one showing that double shuffle elements are determined by less than the full set of stuffle relations, and the other a cyclic property of their coefficients. Although simple, the properties have some useful applications, of which we give two. The first is a generalization of a theorem of Ihara on the abelianizations of elements of the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}$ to elements of the double shuffle Lie algebra in a much larger quotient of the polynomial algebra than the abelianization, namely the trace quotient introduced by Alekseev and Torossian. The second application is a proof that the Grothendieck–Teichmüller Lie algebra $\mathfrak{grt}$ injects into the double shuffle Lie algebra $\mathfrak{ds}$, based on the recent proof by H. Furusho of this theorem in the pro-unipotent situation, but in which the combinatorial properties provide a significant simplification.


Published: 1 January 2012
First available in Project Euclid: 24 October 2018

zbMATH: 1321.17012
MathSciNet: MR3051239

Digital Object Identifier: 10.2969/aspm/06310059

Primary: 05E99, 12Y05, 17B40, 17B65, 17B70

Rights: Copyright © 2012 Mathematical Society of Japan


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