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The Allison–Faulkner construction of $E_8$

Part of: Other nonassociative rings and algebras Linear algebraic groups and related topics Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  10 September 2021

Victor Petrov Open the ORCID record for Victor Petrov [Opens in a new window]  and
Simon W. Rigby Open the ORCID record for Simon W. Rigby [Opens in a new window]
Victor Petrov
Affiliation:
St. Petersburg State University, 29B Line 14th (Vasilyevsky Island), 199178St. Petersburg, Russia PDMI RAS, Nab. Fontanki 27, 191023St. Petersburg, Russia
Simon W. Rigby
Affiliation:
Department of Mathematics, Algebra and Geometry, Ghent University, Krijgslaan 281, 9000Ghent, Belgium e-mail: simon.rigby@ugent.be
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Abstract

We show that the Tits index $E_{8,1}^{133}$ cannot be obtained by means of the Tits construction over a field with no odd degree extensions. The proof uses a general method coming from the theory of symmetric spaces. We construct two cohomological invariants, in degrees $6$ and $8$ , of the Tits construction and the more symmetric Allison–Faulkner construction of Lie algebras of type $E_8$ and show that these invariants can be used to detect the isotropy rank.

Keywords

Exceptional algebraic groupsTits constructionstructurable algebras

MSC classification

Primary: 20G41: Exceptional groups
Secondary: 17B25: Exceptional (super)algebras 17D99: None of the above, but in this section
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 3 , September 2022 , pp. 686 - 701
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

The first author was supported by RFBR grant 19-01-00513. The second author was supported by FWO project G004018N.

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