An explicit classification of dual pairs in exceptional Lie algebras

TL;DR

This paper explicitly classifies all semisimple dual pairs within exceptional Lie algebras, building on Rubenthaler's 1994 outline, and provides detailed examples to enhance understanding and usability.

Contribution

It completes and clarifies the classification of dual pairs in exceptional Lie algebras, making Rubenthaler's 1994 framework more accessible and applicable.

Findings

Complete list of semisimple dual pairs in exceptional Lie algebras

Clarification of methods to verify dual pairs

Numerous examples illustrating the classification

Abstract

The primary goal of this paper is to explicitly write down all semisimple dual pairs in the exceptional Lie algebras. (A dual pair in a reductive Lie algebra $\mathfrak{g}$ is a pair of subalgebras such that each member equals the other's centralizer in $\mathfrak{g}$.) In a 1994 paper, H. Rubenthaler outlined a process for generating a complete list of candidate dual pairs in each of the exceptional Lie algebras. However, the process of checking whether each of these candidate dual pairs is in fact a dual pair is not easy, and requires several distinct insights and methods. In this paper, we carry out this process and explain the relevant concepts as we go. We also give plenty of examples with the hopes of making Rubenthaler's 1994 result not only more complete but more usable and understandable.