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 ${\mathfrak g}\hookrightarrow \mathbb W_n\otimes \operatorname{End}{} V$, 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.