Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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   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 $(G^{\sigma_3} \cap
G^{\tau_3})_0 \subseteq K \subseteq G^{\sigma_3} \cap G^{\tau_3}$.

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The categories of this paper are: atom-ph
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The categories of this paper are: dg-ga
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
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The categories of this paper are: nucl-th
3
The categories of this paper are: physics
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The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
1
The categories of this paper are: cs
4
The categories of this paper are: cond-mat
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The categories of this paper are: q-alg
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph