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

Abstract

R. Lutz introduced the notion of $\Gamma$-symmetric space as a generalization of the classical notion of symmetric space in 1981, where $\Gamma$ 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 $(G^{{\sigma }_3}\cap G^{{\tau }_3}{)}_0\subseteq K\subseteq G^{{\sigma }_3}\cap G^{{\tau }_3}$.

arXiv comment44 pages