Low Cohomogeneity and Polar Actions on Exceptional Compact Lie Groups

TL;DR

This paper classifies and analyzes isometric Lie group actions, especially polar actions, on exceptional compact Lie groups E6, E7, E8, F4, and G2, focusing on low cohomogeneity cases.

Contribution

It provides a comprehensive classification of polar actions and low cohomogeneity isometric actions on these exceptional groups, including principal isotropy algebras.

Findings

Classified all polar actions on exceptional groups.

Determined all low cohomogeneity actions on E6, E7, F4, and E8.

Identified principal isotropy algebras for G2.

Abstract

We study isometric Lie group actions on the compact exceptional groups E6, E7, E8, F4 and G2 endowed with a biinvariant metric. We classify polar actions on these groups. We determine all isometric actions of cohomogeneity less than three on E6, E7, F4 and all isometric actions of cohomogeneity less than 20 on E8. Moreover we determine the principal isotropy algebras for all isometric actions on G2.