The Elliptic Kashiwara-Vergne Lie algebra in low weights

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 $\dim (\mathfrak{krv}{)}^{(3,m)}=\lfloor \frac{m-1}{2}\rfloor -\lfloor \frac{m-1}{3}\rfloor$. In particular, we show that in those degrees there are no odd elements and also confirm Enriquez' conjecture in those degrees.