On the elliptic Kashiwara-Vergne Lie algebra

TL;DR

This paper demonstrates the equivalence of two independently defined elliptic versions of the Kashiwara-Vergne Lie algebra, connecting topological and algebraic approaches in the context of surface fundamental groups.

Contribution

It proves the coincidence of the elliptic Kashiwara-Vergne Lie algebra definitions from topological and algebraic perspectives.

Findings

The Lie algebras rak{krv}^{(1,1)} and rak{krv}_{ell} are shown to be identical.

Establishes a link between topological formality and mould theoretic approaches.

Advances understanding of elliptic Lie algebra structures in mathematical physics.

Abstract

We recall the definitions of two independently defined elliptic versions of the Kashiwara-Vergne Lie algebra $\frak{krv}$, namely the Lie algebra $\frak{krv}^{(1,1)}$ constructed by A.Alekseev, N.Kawazumi, Y.Kuno and F.Naef arising from the study of graded formality isomorphisms associated to topological fundamental groups of surfaces, and the Lie algebra $\frak{krv}_{ell}$ defined using mould theoretic techniques arising from multiple zeta theory by E.Raphael and L.Schneps, and show that they coincide.