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

Abstract

The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{é}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{é}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 ${\tilde{w}}_{{}_4}{\tilde{\upsilon }}_{{}_4}$ and ${\tilde{\mu }}_{{}_4}$ of order four on $E_8$ induced by the $C$-linear transformations $w_{{}_4},{\upsilon }_{{}_4}$ and ${\mu }_{{}_4}$ of the 248-dimensional vector space ${{\mathfrak{e}}_8}^C$, respectively. Further, we determine the structure of these fixed points subgroups $(E_8{)}^{w_{{}_4}},(E_8{)}^{{}_{{\upsilon }_{{}_4}}}$ and $(E_8{)}^{{}_{{\mu }_{{}_4}}}$ 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 $\mathfrak{h}=i\mathbf{R}\oplus \mathfrak{su}(8),i\mathbf{R}\oplus {\mathfrak{e}}_7$ and $\mathfrak{h}=\mathfrak{su}(2)\oplus \mathfrak{su}(8)$, where $\mathfrak{h}=\mathrm{Lie}(H)$.

17 p

References in corpus (1)

• Exceptional Lie groups