Tsukuba Journal of Mathematics

Realizations of inner automorphisms of order 4 and fixed points subgroups by them on the connected compact exceptional lie group $E_8$, Part I

Toshikazu Miyashita

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Abstract

The compact simply connected Riemannian 4-symmetric spaces were classified by J. A. Jiménez as the type of Lie algebra. Needless to say, these spaces as homogeneous manifolds are of the form $G/H$, where $G$ is a connected compact simple Lie group with an automorphism $\tilde{\gamma}$ of order 4 on $G$ and $H$ is a fixed points subgroup $G^\gamma$ of $G$. In the present article, as Part I, for the connected compact exceptional Lie group $E_8$, we give the explicit form of automorphism $\tilde{\sigma}'_4$ of order 4 on $E_8$ induced by the $C$-linear transformation $\sigma'_4$ of 248-dimensional vector space $𝔢^{C}_{8}$ and determine the structure of the group $(E_8)^{\sigma'_4}$. This amounts to the global realization of one of seven cases with an automorphism of order 4 corresponding to the Lie algebra $𝔥 = 𝔰𝔬(6) \oplus 𝔰𝔬(10)$.

Article information

Source
Tsukuba J. Math., Volume 41, Number 1 (2017), 91-166.

Dates
Received: 21 March 2017
Revised: 12 June 2017
First available in Project Euclid: 25 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.tkbjm/1506353561

Digital Object Identifier
doi:10.21099/tkbjm/1506353561

Mathematical Reviews number (MathSciNet)
MR3705776

Zentralblatt MATH identifier
1379.53065

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 4 and fixed points subgroups by them on the connected compact exceptional lie group $E_8$, Part I. Tsukuba J. Math. 41 (2017), no. 1, 91--166. doi:10.21099/tkbjm/1506353561. https://projecteuclid.org/euclid.tkbjm/1506353561


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