## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Galois–Teichmüller Theory and Arithmetic Geometry, H. Nakamura, F. Pop, L. Schneps and A. Tamagawa, eds. (Tokyo: Mathematical Society of Japan, 2012), 59 - 89

### 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**

Received: 12 January 2011

Revised: 10 February 2012

First available in Project Euclid:
24 October 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/aspm/06310059

**Mathematical Reviews number (MathSciNet)**

MR3051239

**Zentralblatt MATH identifier**

1321.17012

**Subjects**

Primary: 17B40: Automorphisms, derivations, other operators 17B65: Infinite-dimensional Lie (super)algebras [See also 22E65] 17B70: Graded Lie (super)algebras 12Y05: Computational aspects of field theory and polynomials 05E99: None of the above, but in this section

**Keywords**

Lie algebras double shuffle multiple zeta values

#### 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