Klein Four subgroups of Lie Algebra Automorphisms

TL;DR

This paper classifies Klein four subgroups within automorphism groups of compact simple Lie algebras, providing a new method for classifying symmetric pairs and spaces with specific involution structures.

Contribution

It introduces a systematic classification of Klein four subgroups in Lie algebra automorphisms, enhancing understanding of symmetric pairs and spaces.

Findings

Classification of Klein four subgroups up to conjugation

Determination of fixed point subgroups for these Klein four subgroups

A new approach to classifying semisimple symmetric pairs

Abstract

By calculating the symmetric subgroups $\Aut(\fru_0)^{\theta}$ and their involution classes, we classify the Klein four subgroups $\Gamma$ of $\Aut(\fru_0)$ for each compact simple Lie algebra $\fru_0$ up to conjugation. This leads to a new approach of classification of semisimple symmetric pairs and $\bbZ_2\times \bbZ_2$-symmetric spaces. We also determine the fixed point subgroup $\Aut(\fru_0)^\Gamma$.