Double Shuffle and Kashiwara-Vergne Lie algebras

TL;DR

This paper establishes an injection from the double shuffle Lie algebra, related to multiple zeta values, into the Kashiwara-Vergne Lie algebra, revealing a deep connection between these algebraic structures.

Contribution

It proves the injectivity of the double shuffle Lie algebra into the Kashiwara-Vergne Lie algebra, using a reformulation and a theorem by Ecalle.

Findings

Double shuffle Lie algebra injects into Kashiwara-Vergne Lie algebra

Reformulation of krv's definition facilitates the proof

Utilizes Ecalle's theorem on properties of ds elements

Abstract

We prove that the double shuffle Lie algebra ds, dual to the space of new formal multiple zeta values, injects into the Kashiwara-Vergne Lie algebra krv defined and studied by Alekseev-Torossian. The proof is based on a reformulation of the definition of krv, and uses a theorem of Ecalle on a property of elements of ds.