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'' ${\mathsf{DMR}}_{\hat{I}}¼(\mathbf{k})$ and to the Betti double shuffle group ${\mathsf{DMR}}^{\mathrm{B}}(\mathbf{k})$ introduced in our earlier work.

160 pages, results on double shuffle schemes and the Betti double shuffle group added