Journal of Lie Theory

no. 4
Structure and Representations for the Electrical Lie Algebra of Type D4
Journal of Lie Theory, Volume 31 (2021) no. 4, pp. 1031-1044
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 $\mathfrak{e}_{D_4}$.
DOI: 10.5802/jolt.1209
Classification: 17B10, 17B20, 17B65, 17B66, 17B68
Keywords: Electrical Lie algebras, 3-step nilpotent Lie algebra, highest weight modules, simple modules 
@article{JOLT_2021_31_4_a9,
     author = {D. Gao and Y. Cai and J. Jiang},
     title = {Structure and {Representations} for the {Electrical} {Lie} {Algebra} of {Type} {D\protect\textsubscript{4}}},
     journal = {Journal of Lie Theory},
     pages = {1031--1044},
     year = {2021},
     volume = {31},
     number = {4},
     doi = {10.5802/jolt.1209},
     zbl = {1490.17009},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1209/}
}
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D. Gao; Y. Cai; J. Jiang. Structure and Representations for the Electrical Lie Algebra of Type D4. Journal of Lie Theory, Volume 31 (2021) no. 4, pp. 1031-1044. doi: 10.5802/jolt.1209

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