Double shuffle relations and double antipodes

Abstract

In this note, we prove that for every Lie series $ψ$ with no terms of degree less than 3, the relation $$[ψ(x,y),x]+[ψ(-x-y,y),-x-y]=0$$ is equivalent to $S_*(ψ_*)=-ψ_*$, where $ψ_*$ denotes the regularization of $ψ$ and $S_*$ is the harmonic antipode. The proof relies on the calculations of the harmonic antipode $S_*$ and the shuffle antipode $S$. As a consequence, we prove that every $ψ$ in Racinet's double shuffle Lie algebra $\mathfrak{dmr}_0$ satisfies the relation $[ψ(x,y),x]+[ψ(-x-y,y),-x-y]=0$. We further prove that $\mathfrak{dmr}_0$ injects into the symmetric Kashiwara--Vergne Lie algebra $\mathfrak{krv}^{\mathrm{sym}}_2$ of Alekseev and Torossian.