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July 2017 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
Tsukuba J. Math. 41(1): 91-166 (July 2017). DOI: 10.21099/tkbjm/1506353561

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 $𝔥 = 𝔰𝔬(6) \oplus 𝔰𝔬(10)$.

Citation

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Toshikazu Miyashita. "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 (1) 91 - 166, July 2017. https://doi.org/10.21099/tkbjm/1506353561

Information

Received: 21 March 2017; Revised: 12 June 2017; Published: July 2017
First available in Project Euclid: 25 September 2017

zbMATH: 1379.53065
MathSciNet: MR3705776
Digital Object Identifier: 10.21099/tkbjm/1506353561

Subjects:
Primary: 17B40, 53C30, 53C35

Rights: Copyright © 2017 University of Tsukuba, Institute of Mathematics

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Vol.41 • No. 1 • July 2017
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