Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 31 (2021), No. 2, 583--598Copyright Heldermann Verlag 2021 The Elliptic Kashiwara-Vergne Lie Algebra in Low Weights Florian Naef School of Mathematics, Trinity College, Dublin, Ireland naeff@tcd.ie Yuting Qin Massachusetts Institute of Technology, Cambridge, U.S.A. emmaqin@mit.edu [Abstract-pdf] We study the elliptic Kashiwara-Vergne Lie algebra $\mathfrak{krv}$, which is a certain Lie sub\-al\-gebra of the Lie algebra of derivations of the free Lie algebra in two generators. It has a na\-tu\-ral bi\-gra\-ding, 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 for $j$ odd. We also compute $$\operatorname{dim}(\mathfrak{krv})^{(3,j)} = \lfloor\frac{j-1}{2}\rfloor - \lfloor\frac{j-1}{3}\rfloor.$$ In particular, we show that in those degrees there are no odd elements and also confirm Enriquez' conjecture in those degrees. Keywords: Elliptic Kashiwara-Vergne Lie algebra. MSC: 17B01. [ Fulltext-pdf  (246  KB)] for subscribers only.