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Ternary vector cross products

R Shaw

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Ternary vector cross products are studied in their own right. Results include a new proof of Hurwitz's theorem and a 'principle of duplicity'. Upon breaking the symmetry in eight dimensions, by choosing a preferred axis, this last principle implies the well known triality principle for octonions and SO(8) transformations. In displaying canonical forms it helps to put the eight basis vectors in correspondence with the eight points of the three-dimensional affine geometry over F2.


PACS

02.20.Qs General properties, structure, and representation of Lie groups

02.10.Ud Linear algebra

02.20.Sv Lie algebras of Lie groups

MSC

15A63 Quadratic and bilinear forms, inner products [See mainly 11Exx]

15A21 Canonical forms, reductions, classification

22E60 Lie algebras of Lie groups (For the algebraic theory of Lie algebras, see 17Bxx)

14Rxx Affine geometry

Subjects

Mathematical physics

Dates

Issue 11 (1 August 1987)



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