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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

## Abstract

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.

## Information

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

zbMATH: 1321.17012
MathSciNet: MR3051239

Digital Object Identifier: 10.2969/aspm/06310059

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