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 ${\frak{ds}}$ into the Kashiwara-Vergne Lie algebra ${\frak{krv}}$, forming a commutative triangle with the known Lie injections of the Grothendieck-Teichm\"uller Lie algebra ${\frak{grt}}\rightarrow {\frak{ds}}$ and ${\frak{grt}}\rightarrow{\frak{krv}}$.