The Elliptic Kashiwara-Vergne Lie algebra in low weights

TL;DR

This paper investigates the structure of the elliptic Kashiwara-Vergne Lie algebra, revealing its low-degree components, confirming a conjecture, and providing explicit dimension formulas for certain graded parts.

Contribution

It provides a detailed analysis of low weight elements of the elliptic Kashiwara-Vergne Lie algebra, including dimension calculations and verification of Enriquez's conjecture.

Findings

vens  elements in v imensional components

Explicit dimension formulas for v imensional components

Confirmation of Enriquez's conjecture in studied degrees

Abstract

In this paper, we study the elliptic Kashiwara-Vergne Lie Algebra $\mathfrak{krv}$, which is a certain Lie subalgebra of the Lie algebra of derivations of the free Lie algebra in two generators. It has a natural bigrading, such that the Lie bracket is of bidegree $(-1,-1)$. After recalling the graphical interpretation of this Lie algebra, we examine low degree elements of $\mathfrak{krv}$. More precisely, we find that $\mathfrak{krv}^{(2,j)}$ is one-dimensional for even $j$ and zero $j$ odd. We also compute $\operatorname{dim}(\mathfrak{krv})^{(3,m)} = \lfloor\frac{m-1}{2}\rfloor - \lfloor\frac{m-1}{3}\rfloor$. In particular, we show that in those degrees there are no odd elements and also confirm Enriquez' conjecture in those degrees.