On some realizations of globally exceptional $\varmathbb{Z}_3 \times \varmathbb{Z}_3 $-symmetric spaces $G/K$, $G=G_2, F_4, E_6$, Part I

TL;DR

This paper constructs and classifies certain exceptional symmetric spaces associated with the groups G2, F4, and E6, using automorphisms of order 3 to realize $ ext{Z}_3 imes ext{Z}_3$-symmetric spaces.

Contribution

It explicitly constructs automorphisms of order 3 on exceptional Lie groups and determines the structure of their fixed point intersections, providing global realizations of these symmetric spaces.

Findings

Explicit automorphisms of order 3 on G2, F4, E6

Determination of the structure of fixed point intersections

Realization of $ ext{Z}_3 imes ext{Z}_3$-symmetric spaces

Abstract

R. Lutz introduced the notion of $\varGamma$-symmetric space as a generalization of the classical notion of symmetric space in 1981, where $\varGamma$ is a finite abelian group. In the present article, as $\varGamma=\varmathbb{Z}_3 \times \varmathbb{Z}_3$, we give the automorphisms $\tilde{\sigma}_3, \tilde{\tau}_3$ of order $3$ on the connected compact exceptional Lie groups $G=G_2, F_4,E_6$ %and construct $\varGamma=\varmathbb{Z}_3 \times \varmathbb{Z}_3$ as the elements of order $3$ in $\Aut(G)$, explicitly and determine the structure of the group $G^{\sigma_3} \cap G^{\tau_3}$ using homomorphism theorem elementary. These amount to some global realizations of exceptional $\varmathbb{Z}_3 \times \varmathbb{Z}_3$-symmetric spaces $G/K$, where $(G^{\sigma_3} \cap G^{\tau_3})_0 \subseteq K \subseteq G^{\sigma_3} \cap G^{\tau_3}$.