Journal of Lie Theory

no. 2
Computing Parabolically Induced Embeddings of Semisimple Complex Lie Algebras in Weyl Algebras
Journal of Lie Theory, Volume 25 (2015) no. 2, pp. 559-577
\def\g{{\frak g}} \def\p{{\frak p}} \def\End{\mathop{\rm End}\nolimits} An arbitrary proper parabolic subalgebra $\p$ of a simple complex Lie algebra $\g$ induces an embedding $\g\to\Bbb W_n$, and more generally an embedding $\g\to\Bbb W_n\otimes \End V$, where $\Bbb W_n$ is the Weyl algebra in $n$ variables, $n$ is the dimension of the nilradical of $\p$, and $V$ is an arbitrary $\p$-module. We give an elementary proof of this known fact, report on a computer program computing the embeddings, and tabulate exceptional Lie algebra embeddings $G_2 \to \Bbb W_5$, $F_4 \to \Bbb W_{15}$, $E_6 \to \Bbb W_{16}$, $E_7 \to\Bbb W_{27}$, $E_8 \to \Bbb W_{57}$ arising in this fashion.
DOI: 10.5802/jolt.851
Classification: 17B20, 17B25, 17B35, 17B66
Keywords: Generalized Verma modules, exceptional Lie algebras, realization of exceptional Lie algebra, Weyl algebra 
@article{JOLT_2015_25_2_a13,
     author = {T. Milev},
     title = {Computing {Parabolically} {Induced} {Embeddings} of {Semisimple} {Complex} {Lie} {Algebras} in {Weyl} {Algebras}},
     journal = {Journal of Lie Theory},
     pages = {559--577},
     year = {2015},
     volume = {25},
     number = {2},
     doi = {10.5802/jolt.851},
     zbl = {1368.17012},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.851/}
}
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T. Milev. Computing Parabolically Induced Embeddings of Semisimple Complex Lie Algebras in Weyl Algebras. Journal of Lie Theory, Volume 25 (2015) no. 2, pp. 559-577. doi: 10.5802/jolt.851

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