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

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-Teichmüller 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 $\mathfrak{krv}\to {\mathfrak{krv}}_{ell}$ extending the known morphisms $\mathfrak{grt}\hookrightarrow {\mathfrak{grt}}_{ell}$ (Enriquez section) and $\mathfrak{ds}\to {\mathfrak{ds}}_{ell}$ (Écalle map).

67 pages