Geometric characterisation of subvarieties of 𝓔6(𝕂) related to the ternions and sextonions

The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety 𝓔 (𝕂) over an arbitrary field 𝕂. The characterised varieties arise as Veronese representations of certain ring projective planes over quadratic subalgebras of the split octonions 𝕆’ over...

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Bibliographic Details
Published inAdvances in geometry Vol. 23; no. 1; pp. 69 - 106
Main Author De Schepper, Anneleen
Format Journal Article
LanguageEnglish
Published De Gruyter 27.01.2023
Subjects
14N05
51A45
51B25
51C05
51E24
composition algebras
Freudenthal–Tits magic square
ring geometries
Veronese varieties
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Summary:The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety 𝓔 (𝕂) over an arbitrary field 𝕂. The characterised varieties arise as Veronese representations of certain ring projective planes over quadratic subalgebras of the split octonions 𝕆’ 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:1615-715X
1615-7168
DOI:10.1515/advgeom-2022-0005