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Algebra |
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Authors: Eckhard Hitzer
We study the embedding of octonions in the Clifford geometric algebra for spacetime STA Cl(1,3), as suggested by Anthony Lasenby at AGACSE 2021. As far as possible, we extend the approach to similar octonion embeddings for all three- and four dimensional Clifford geometric algebras Cl(p,q), n = p + q = 3,4. Noticeably, the lack of a quaternionic subalgebra in Cl(2,1), seems to prevent the construction of an octonion embedding in this case, and necessitates a special approach in Cl(2,2). As examples, we present for Cl(3,0) the non-associativity of the octonionic product in terms of multivector grade parts with cyclic symmetry, show how octonion products and involutions can be combined to make the opposite transition from octonions to the Clifford geometric algebra Cl(3,0), and how octonionic multiplication can be represented with (complex) biquaternions or Pauli matrix algebra.
Comments: 24 Pages. 6 figures, 7 tables.
Download: PDF
[v1] 2022-02-16 07:20:31
Unique-IP document downloads: 581 times
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