Computing parabolically induced embeddings of semisimple complex Lie algebras in Weyl algebras

Abstract

An arbitrary proper parabolic subalgebra $\mathfrak{p}$ of a simple complex Lie algebra $\mathfrak{g}$ induces an embedding $\mathfrak{g}\hookrightarrow {\mathbb{W}}_n$, and more generally an embedding $\mathfrak{g}\hookrightarrow {\mathbb{W}}_n\otimes \mathrm{End}V$, where ${\mathbb{W}}_n$ is the Weyl algebra in $n$ variables, $n$ is the dimension of the nilradical of $\mathfrak{p}$, and $V$ is an arbitrary $\mathfrak{p}$-module. We give an elementary proof of this known fact, report on a computer program computing the embeddings, and tabulate exceptional Lie algebra embeddings $G_2\hookrightarrow {\mathbb{W}}_5$, $F_4\hookrightarrow {\mathbb{W}}_{15}$, $E_6\hookrightarrow {\mathbb{W}}_{16}$, $E_7\hookrightarrow {\mathbb{W}}_{27}$, $E_8\hookrightarrow {\mathbb{W}}_{57}$ arising in this fashion.