The double shuffle Lie algebra injects into the Kashiwara-Vergne Lie algebra

Abstract

In this article we prove that there exists an injective Lie morphism from the double shuffle Lie algebra $\mathfrak{ds}$ into the Kashiwara-Vergne Lie algebra $\mathfrak{krv}$, forming a commutative triangle with the known Lie injections of the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}\to \mathfrak{ds}$ and $\mathfrak{grt}\to \mathfrak{krv}$.