Double shuffle Lie algebra and special derivations

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.