
Journal of Lie Theory 31 (2021), No. 4, 10311044 Copyright Heldermann Verlag 2021 Structure and Representations for the Electrical Lie Algebra of Type D_{4} Dongfang Gao School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, P. R. China gaodfw@mail.ustc.edu.cn Yanan 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 [Abstractpdf] 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, 3step nilpotent Lie algebra, highest weight modules, simple modules. MSC: 17B10, 17B20, 17B65, 17B66, 17B68. [ Fulltextpdf (153 KB)] for subscribers only. 