
Journal of Lie Theory 31 (2021), No. 2, 583598 Copyright Heldermann Verlag 2021 The Elliptic KashiwaraVergne 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 [Abstractpdf] We study the elliptic KashiwaraVergne 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 onedimensional for even $j$ and zero for $j$ odd. We also compute $$ \operatorname{dim}(\mathfrak{krv})^{(3,j)} = \lfloor\frac{j1}{2}\rfloor  \lfloor\frac{j1}{3}\rfloor. $$ In particular, we show that in those degrees there are no odd elements and also confirm Enriquez' conjecture in those degrees. Keywords: Elliptic KashiwaraVergne Lie algebra. MSC: 17B01. [ Fulltextpdf (246 KB)] for subscribers only. 