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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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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Abstract 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.
AbstractList 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.
Author De Schepper, Anneleen
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Quaternions
Title A geometric characterisation of subvarieties of the standard E_6-variety related to the ternions, degenerate split quaternions and sextonions over arbitrary fields
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