Journal of Lie Theory 31 (2021), No. 4, 1031--1044
Copyright Heldermann Verlag 2021
Structure and Representations for the Electrical Lie Algebra of Type D4
Dongfang Gao
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, P. R. China
gaodfw@mail.ustc.edu.cn
Yan-an Cai
Dept. of Mathematics, Soochow University, Suzhou, Jiangsu, P. R. China
yatsai@mail.ustc.edu.cn
Jin Jiang
Dept. of Mathematics, Soochow University, Suzhou, Jiangsu, P. R. China
mxsl369@163.com
[Abstract-pdf]
We prove the dimension conjecture for electrical Lie algebra $\mathfrak{e}_{D_4}$ of type $D_4$. Moreover, we present a new method to construct $3$-step nilpotent Lie algebras and show that $\mathfrak{e}_{D_4}$ is isomorphic to the semidirect product of $\mathfrak{s}\mathfrak{l}_2$ with a $3$-step nilpotent Lie algebra constructed from the colored complete bipartible graph $K_{2,2}$. Also, we classify all simple highest weight modules for $\mathfrak{e}_{D_4}$.
Keywords: Electrical Lie algebras, 3-step nilpotent Lie algebra, highest weight modules, simple modules.
MSC: 17B10, 17B20, 17B65, 17B66, 17B68.
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