Brunnian braids and the inclusion from double shuffle Lie algebra to Kashiwara-Vergne Lie algebra

TL;DR

This paper proves an injection of the double shuffle Lie algebra into the symmetric Kashiwara-Vergne Lie algebra using brunnian braid groups, offering a new approach beyond mould calculus and generalizing the inclusion.

Contribution

It introduces a novel proof of the injection using brunnian braid groups and extends the inclusion via lower central series and explicit algebraic links.

Findings

Injection of rak{dmr}_0 into rak{krv}^{sym}_2 established

New proof method based on brunnian braid groups

Generalizations linking pentagon, stuffle coproduct, divergence, and necklace cobracket

Abstract

It is proved by L.~Schneps that the double shuffle Lie algebra $\mathfrak{dmr}_0$ injects to the Kashiwara-Vergne Lie algebra $\mathfrak{krv}_2$ in \cite{Schneps2012,Schneps2025}. We show that $\mathfrak{dmr}_0$ with the infinitesimal hexagon equation $[x,\varphi(-x,-y,x)]+[y,\varphi(-x-y,y)]=0$ injects to the symmetric Kashiwara-Vergne Lie algebra $\mathfrak{krv}^{\mathrm{sym}}_2$. The proof is based on the inclusion of brunnian braids group on different genus 0 surfaces which is different from the method of mould calculus in \cite{Schneps2012,Schneps2025}. We generalize the inclusion in two directions, one using lower central series of brunnian Lie algebras and the other is to establish explicit links between the pentagon equation map, the stuffle coproduct, the divergence map and the necklace cobracket.