On linearised and elliptic versions of the Kashiwara-Vergne Lie algebra
Hidekazu Furusho, Nao Komiyama, Elise Raphael, Leila Schneps
TL;DR
The paper develops two new LG-algebraic avatars of the Kashiwara–Vergne framework: a depth-graded linearized version $\mathfrak{lkrv}$ and an elliptic version $\mathfrak{krv}_{ell}$, and situates them within the broader context of elliptic and non-elliptic analogues such as $\mathfrak{grt}$, $\mathfrak{ds}$ and their elliptic counterparts. Using Écalle’s mould theory, it shows that depth-graded and elliptic constructions share identical defining properties, enabling explicit injections: $gr\,\mathfrak{krv} \hookrightarrow \mathfrak{lkrv}$ and $\mathfrak{ls} \hookrightarrow \mathfrak{lkrv}$, with depth-3 coincidences providing concrete dimensions. It further defines $\mathfrak{krv}_{ell}$ as a subspace of push-invariant, circ-neutral mould data, proves it is a Lie algebra and constructs a canonical injection $\mathfrak{krv} \hookrightarrow \mathfrak{krv}_{ell}$ via a four-step mould-theoretic process, including the crucial map $\Xi$. The work also clarifies how elliptic versions of grt and ds relate within commutative diagrams, including a subalgebra $\widetilde{\mathfrak{grt}}_{ell}$ to bridge gaps in known injections, and highlights mould-theoretic identities (fundamental identity) as central to these connections. Overall, the results solidify a parallel between linearized/depth-graded and elliptic versions of krv, ds, and grt and provide concrete algebraic tools for comparing their structures and graded components.
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 $\mathfrak{krv}\rightarrow \mathfrak{krv}_{ell}$ extending the known morphisms $\mathfrak{grt}\hookrightarrow\mathfrak{grt}_{ell}$ (Enriquez section) and $\mathfrak{ds}\rightarrow\mathfrak{ds}_{ell}$ (Écalle map).