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.
1
School of Mathematics, Trinity College, Dublin, Ireland
2
Massachusetts Institute of Technology, Cambridge, U.S.A.
Florian Naef; Yuting Qin. The Elliptic Kashiwara-Vergne Lie Algebra in Low Weights. Journal of Lie Theory, Tome 31 (2021) no. 2, pp. 583-598. http://geodesic.mathdoc.fr/item/JOLT_2021_31_2_a15/
@article{JOLT_2021_31_2_a15,
author = {Florian Naef and Yuting Qin},
title = {The {Elliptic} {Kashiwara-Vergne} {Lie} {Algebra} in {Low} {Weights}},
journal = {Journal of Lie Theory},
pages = {583--598},
year = {2021},
volume = {31},
number = {2},
url = {http://geodesic.mathdoc.fr/item/JOLT_2021_31_2_a15/}
}
TY - JOUR
AU - Florian Naef
AU - Yuting Qin
TI - The Elliptic Kashiwara-Vergne Lie Algebra in Low Weights
JO - Journal of Lie Theory
PY - 2021
SP - 583
EP - 598
VL - 31
IS - 2
UR - http://geodesic.mathdoc.fr/item/JOLT_2021_31_2_a15/
ID - JOLT_2021_31_2_a15
ER -
%0 Journal Article
%A Florian Naef
%A Yuting Qin
%T The Elliptic Kashiwara-Vergne Lie Algebra in Low Weights
%J Journal of Lie Theory
%D 2021
%P 583-598
%V 31
%N 2
%U http://geodesic.mathdoc.fr/item/JOLT_2021_31_2_a15/
%F JOLT_2021_31_2_a15
