The Elliptic Kashiwara-Vergne Lie Algebra in Low Weights

We study the elliptic Kashiwara-Vergne Lie algebra 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 krv. More precisely, we find that krv((2,j)) is one-dimensional for even j and zero for j odd. We also compute dim(krv)((3,j)) = [j-1/2] - [j-1/3]. In particular, we show that in those degrees there are no odd elements and also confirm Enriquez' conjecture in those degrees.