## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 43, Number 1 (2019), 1-22.

### 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. Jiménez 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. Jiménez, 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, 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 $𝔢^C_8$, 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 $𝔥 = i \boldsymbol{R} \oplus 𝔰𝔲(8)$, $i\boldsymbol{R} \oplus 𝔢_7$ and $𝔥 = 𝔰𝔲(2) \oplus 𝔰𝔲(8)$, where $𝔥 = \mathrm{Lie}(H)$.

#### Article information

**Source**

Tsukuba J. Math., Volume 43, Number 1 (2019), 1-22.

**Dates**

First available in Project Euclid: 25 October 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1571968818

**Digital Object Identifier**

doi:10.21099/tkbjm/1571968818

**Mathematical Reviews number (MathSciNet)**

MR4023311

**Subjects**

Primary: 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15] 53C35: Symmetric spaces [See also 32M15, 57T15] 17B40: Automorphisms, derivations, other operators

**Keywords**

4-symmetric spaces exceptional Lie groups

#### Citation

Miyashita, Toshikazu. Realizations of inner automorphisms of order four and fixed points subgroups by them on the connected compact exceptional Lie group $E_8$, Part II. Tsukuba J. Math. 43 (2019), no. 1, 1--22. doi:10.21099/tkbjm/1571968818. https://projecteuclid.org/euclid.tkbjm/1571968818