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

Todor Milev¹

¹Dept. of Mathematics, University of Massachusetts, 100 William T. Morrissey Blvd, Boston, MA 02125, U.S.A.

Journal of Lie Theory, Tome 25 (2015) no. 2, pp. 559-577

Résumé

\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.

Détail

Classification : 17B20, 17B25, 17B35, 17B66
Mots-clés : Generalized Verma modules, exceptional Lie algebras, realization of exceptional Lie algebra, Weyl algebra