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

TL;DR

This paper presents a straightforward proof and computational methods for embedding semisimple complex Lie algebras into Weyl algebras via parabolic subalgebras, including explicit embeddings for exceptional Lie algebras.

Contribution

It provides an elementary proof of embeddings of Lie algebras into Weyl algebras and tabulates explicit embeddings for exceptional Lie algebras.

Findings

Elementary proof of Lie algebra embeddings into Weyl algebras.

Computer program for computing these embeddings.

Tabulated embeddings for exceptional Lie 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.