July 2022 Realizations of inner automorphisms of order four and fixed points subgroups by them on the connected compact exceptional Lie group E8, Part III
Toshikazu Miyashita
Tsukuba J. Math. 46(1): 39-65 (July 2022). DOI: 10.21099/tkbjm/20224601039

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 γ˜ of order four on G and H is a fixed points subgroup Gγ 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 E8. In the present article, we give the explicit form of automorphisms ω˜4, κ˜4 and ε˜4 of order four on E8 induced by the C-linear transformations ω4, κ4 and ε4 of the 248-dimensional C-vector space e8C, respectively. Further, we determine the structure of these fixed points subgroups (E8)ω4, (E8)κ4 and (E8)ε4 of E8. These amount to the global realizations of three spaces among seven Riemannian 4-symmetric spaces G/H above corresponding to the Lie algebras h=su(2)iRe6, iRso(14) and h=su(2)iRso(12), where h=Lie(H). With this article, the all realizations of inner automorphisms of order four and fixed points subgroups by them have been completed in E8.

Citation

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Toshikazu Miyashita. "Realizations of inner automorphisms of order four and fixed points subgroups by them on the connected compact exceptional Lie group E8, Part III." Tsukuba J. Math. 46 (1) 39 - 65, July 2022. https://doi.org/10.21099/tkbjm/20224601039

Information

Received: 19 November 2020; Revised: 28 October 2021; Published: July 2022
First available in Project Euclid: 19 September 2022

MathSciNet: MR4489187
zbMATH: 1503.53109
Digital Object Identifier: 10.21099/tkbjm/20224601039

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

Keywords: 4-symmetric spaces , exceptional Lie groups

Rights: Copyright © 2022 University of Tsukuba, Institute of Mathematics

Vol.46 • No. 1 • July 2022
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