On linearised and elliptic versions of the Kashiwara-Vergne Lie algebra

TL;DR

This paper introduces linearized and elliptic variants of the Kashiwara-Vergne Lie algebra, exploring their relationships with elliptic versions of other significant Lie algebras in mathematical physics and number theory.

Contribution

It defines the linearized and elliptic Kashiwara-Vergne Lie algebras and establishes their connections with elliptic versions of the Grothendieck-Teichmüller and double shuffle Lie algebras.

Findings

Defined the linearized Kashiwara-Vergne Lie algebra $raklkv$.

Constructed the elliptic Kashiwara-Vergne Lie algebra $rakkrv_{ell}$.

Established injective Lie algebra morphisms linking these new algebras to existing elliptic structures.

Abstract

The goal of this article is to define a linearized or depth-graded version $\mathfrak{lkv}$, and a closely related elliptic version $\mathfrak{krv}_{ell}$, of the Kashiwara-Vergne Lie algebra $\mathfrak{krv}$ originally constructed by Alekseev and Torossian as the space of solutions to the linearized Kashiwara-Vergne problem. We show how the elliptic Lie algebra $\mathfrak{krv}_{ell}$ is related to earlier constructions of elliptic versions $\mathfrak{grt}_{ell}$ and $\mathfrak{ds}_{ell}$ of the Grothendieck-Teichm\"uller Lie algebra $\mathfrak{grt}$ and the double shuffle Lie algebra $\mathfrak{ds}$. In particular we show that there is an injective Lie morphism $\mathfrak{ds}_{ell}\hookrightarrow \mathfrak{krv}_{ell}$, and an injective Lie algebra morphism $\mathfrak{krv}\rightarrow \mathfrak{krv}_{ell}$ extending the known morphisms $\mathfrak{grt}\hookrightarrow\mathfrak{grt}_{ell}$ (Enriquez section) and $\mathfrak{ds}\rightarrow\mathfrak{ds}_{ell}$ (\'Ecalle map).