Double shuffle relations and double antipodes

Abstract

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

12 pages, corrected some typos and improved the language, submitted version