A geometric characterisation of subvarieties of the standard E_6-variety related to the ternions, degenerate split quaternions and sextonions over arbitrary fields

The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety (the standard projective variety associated to the split exceptional group of Lie type E_6) over an arbitrary field K. The characterised varieties arise as Veronese representations of cer...

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Bibliographic Details
Published inarXiv.org
Main Author De Schepper, Anneleen
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 09.06.2020
Subjects
Algebra
Quaternions
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Summary:The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety (the standard projective variety associated to the split exceptional group of Lie type E_6) over an arbitrary field K. The characterised varieties arise as Veronese representations of certain ring projective planes over quadratic subalgebras of the split octonions over K (among which the sextonions, a 6-dimensional non-associative algebra). We describe how these varieties are linked to the Freudenthal-Tits magic square, and discuss how they would even fit in, when also allowing the sextonions and other "degenerate composition algebras" as the algebras used to construct the square.
ISSN:2331-8422