The Elliptic Kashiwara-Vergne Lie Algebra in Low Weights
Journal of Lie Theory, Volume 31 (2021) no. 2, pp. 583-598
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.
@article{JOLT_2021_31_2_a15,
author = {F. Naef and Y. 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},
doi = {10.5802/jolt.1187},
zbl = {1482.17017},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1187/}
}
F. Naef; Y. Qin. The Elliptic Kashiwara-Vergne Lie Algebra in Low Weights. Journal of Lie Theory, Volume 31 (2021) no. 2, pp. 583-598. doi: 10.5802/jolt.1187
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