关于全局例外$\varmathbb{Z}_3 \times \varmathbb{Z}_3 $ -对称空间$G/K$，$G=G_2, F_4, E_6$的几种实现--第I部分
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
摘要：
R. Lutz在1981年引入了$\varGamma$ -对称空间的概念，作为对称空间的经典概念的推广，其中$\varGamma$是有限阿贝尔群。作为$\varGamma=\varmathbb{Z}_3 \times \varmathbb{Z}_3$，给出了连通紧致例外李群$G=G_2, F_4,E_6$ %上的$3$阶自同构$\tildeσ_3, \tildeτ_3$，并构造了$\varGamma=\varmathbb{Z}_3 \times \varmathbb{Z}_3$作为$\Aut(G)$中的$3$阶元素，明确地利用同态定理初等地确定了群$G^{σ_3} \cap G^{τ_3}$的结构.这相当于特殊$\varmathbb{Z}_3 \times \varmathbb{Z}_3$ -对称空间$G/K$的某些全局实现，其中$(G^{σ_3} \cap G^{τ_3})_0 \subseteq K \subseteq G^{σ_3} \cap G^{τ_3}$ .
英文摘要：
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 $\tildeσ_3, \tildeτ_3$ 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 $G^{σ_3} \cap G^{τ_3}$ using homomorphism theorem elementary. These amount to some global realizations of exceptional $\varmathbb{Z}_3 \times \varmathbb{Z}_3$-symmetric spaces $G/K$, where $(G^{σ_3} \cap G^{τ_3})_0 \subseteq K \subseteq G^{σ_3} \cap G^{τ_3}$.
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