Realizations of inner automorphisms of order four and fixed points subgroups by them on the connected compact exceptional Lie group $E_8$, Part III

TL;DR

This paper explicitly constructs automorphisms of order four on the exceptional Lie group E8 and determines their fixed point subgroups, completing the classification of such automorphisms and associated symmetric spaces.

Contribution

It provides explicit forms of order four automorphisms on E8 and characterizes their fixed point subgroups, advancing the understanding of 4-symmetric spaces related to E8.

Findings

Explicit automorphisms of order four on E8 are given.

Fixed point subgroups of these automorphisms are determined.

Classification of all such automorphisms and subgroups in E8 is completed.

Abstract

The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{\'{e}}nez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $\tilde{\gamma}$ of order four on $G$ and $H$ is a fixed points subgroup $G^\gamma$ of $G$. According to the classification by J.A. Jim{\'{e}}nez, there exist seven compact simply connected Riemannian 4-symmetric spaces $ G/H $ in the case where $ G $ is of type $ E_8 $. In the present article, %as Part II continuing from Part I, for the connected compact %exceptional Lie group $E_8$, we give the explicit form of automorphisms $\tilde{\omega}^{}_4,\tilde{\kappa}^{}_4$ and $ \tilde{\varepsilon}^{}_4$ of order four on $E_8$ induced by the $C$-linear transformations $\omega^{}_4, \kappa^{}_4$ and $ \varepsilon^{}_4$ of the 248-dimensional $ C $-vector space ${\mathfrak{e}_8}^{C}$, respectively. Further, we determine the structure of these fixed points subgroups $(E_8)^{{}_{\omega^{}_4}}, (E_8)^{{}_{\kappa^{}_4}}$ and $(E_8)^{{}_{\varepsilon^{}_4}}$ of $ E_8 $. These amount to the global realizations of three spaces among seven Riemannian 4-symmetric spaces $ G/H $ above corresponding to the Lie algebras $ \mathfrak{h}=\mathfrak{su}(2)\oplus i\mathbf{R} \oplus \mathfrak{e}_6, i\mathbf{R} \oplus \mathfrak{so}(14)$ and $\mathfrak{h} =\mathfrak{su}(2) \oplus i\mathbf{R}\oplus \mathfrak{so}(12)$, where $ \mathfrak{h}={\rm Lie}(H) $. With this article, the all realizations of inner automorphisms of order four and fixed points subgroups by them have been completed in $ E_8 $.