Notes on Kashiwara-Vergne and double shuffle Lie algebras

TL;DR

This paper explores the relationship between the Kashiwara-Vergne and double shuffle Lie algebras, and verifies Ecalle's senary relation for low depths, advancing understanding in algebraic structures related to multiple zeta values.

Contribution

It clarifies the connection between two important Lie algebras and confirms a key relation in specific cases, contributing to the theoretical framework.

Findings

Confirmed Ecalle's senary relation for small depths

Explained the relationship between kishara-Vergne and double shuffle Lie algebras

Provided insights into algebraic structures related to multiple zeta values

Abstract

We explain the current situation of the relationship between the Kashiwara-Vergne Lie algebra $\mathfrak{krv}$ and the double shuffle Lie algebra $\mathfrak{dmr}$. We also show the validity of Ecalle's senary relation for small depths.