## Advanced Studies in Pure Mathematics

### Combinatorics of the double shuffle Lie algebra

#### 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.

#### Article information

Dates
Revised: 10 February 2012
First available in Project Euclid: 24 October 2018

https://projecteuclid.org/ euclid.aspm/1540417814

Digital Object Identifier
doi:10.2969/aspm/06310059

Mathematical Reviews number (MathSciNet)
MR3051239

Zentralblatt MATH identifier
1321.17012

#### Citation

Carr, Sarah; Schneps, Leila. Combinatorics of the double shuffle Lie algebra. Galois–Teichmüller Theory and Arithmetic Geometry, 59--89, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06310059. https://projecteuclid.org/euclid.aspm/1540417814