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Journal of Lie Theory 25 (2015), No. 2, 559--577
Copyright Heldermann Verlag 2015

Computing Parabolically Induced Embeddings of Semisimple Complex Lie Algebras in Weyl Algebras

Todor Milev
Dept. of Mathematics, University of Massachusetts, 100 William T. Morrissey Blvd, Boston, MA 02125, U.S.A.


\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.

Keywords: Generalized Verma modules, exceptional Lie algebras, realization of exceptional Lie algebra, Weyl algebra.

MSC: 17B20, 17B25, 17B35, 17B66

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