Double shuffle Lie algebra and special derivations

TL;DR

This paper investigates the structure of Racinet's double shuffle Lie algebra, proving its containment in a subalgebra of special derivations, its stability under an involution, and deriving related algebraic relations and inclusions.

Contribution

It establishes the inclusion of the double shuffle Lie algebra in the special derivations subalgebra and proves its stability under an involution, extending previous conditional results.

Findings

$rak{dmr}_0$ is contained in $rak{sder}$.

$rak{dmr}_0$ is stable under an involution exchanging $x_0$ and $x_ty$.

$rak{dmr}_0$ satisfies the senary relation and is included in $rak{krv}_2$.

Abstract

Racinet's double shuffle Lie algebra $\mathfrak{dmr}_0$ is a Lie subalgebra of the Lie algebra $\mathfrak{tder}$ of tangential derivations of the free Lie algebra with generators $x_0,x_1$, i.e. of derivations such that $x_1\mapsto 0$ and $x_0\mapsto [a,x_0]$ for some element $a$. We prove: (1) $\mathfrak{dmr}_0$ is contained in the Lie subalgebra $\mathfrak{sder}$ of $\mathfrak{tder}$ of special derivations, i.e. satisfying the additional condition that $x_\infty\mapsto [b,x_\infty]$ for some element $b$, where $x_\infty:=x_1-x_0$; (2) $\mathfrak{dmr}_0$ is stable under the involution of $\mathfrak{sder}$ induced by the exchange of $x_0$ and $x_\infty$. The first statement: (a) says that any element of $\mathfrak{dmr}_0$ satisfies the "senary relation" (a fact announced without proof by Ecalle in 2011); (b) implies the inclusion $\mathfrak{dmr}_0\subset \mathfrak{krv}_2$ (which was proved by Schneps in 2012 only conditionally to the truth of (1)). We also derive the analogues of statements (a) and (b) respective to Racinet's ``double shuffle schemes'' $\mathsf{DMR}_\mu(\mathbf k)$ and to the Betti double shuffle group $\mathsf{DMR}^{\mathrm{B}}(\mathbf k)$ introduced in our earlier work.