Clifford algebra-parametrized octonions and generalizations

Abstract

Introducing products between multivectors of Cl(0,7) and octonions, resulting in an octonion, and leading to the non-associative standard octonionic product in a particular case, we generalize the octonionic X-product, associated with the transformation rules for bosonic and fermionic fields on the tangent bundle over the 7-sphere, and the XY-product. We also present the formalism necessary to construct Clifford algebra-parametrized octonions. Finally we introduce a method to construct generalized octonionic algebras, where their octonionic units are parametrized by arbitrary Clifford multivectors. The products between Clifford multivectors and octonions, leading to an octonion, are shown to share graded-associative, supersymmetric properties. We also investigate the generalization of Moufang identities, for each one of the products introduced. The X-product equals twice the parallelizing torsion, given by the torsion tensor, and is used to investigate the S7 Kac-Moody algebra. The X-product has also been used to obtain triality maps and G2 actions, and it leads naturally to remarkable geometric and topological properties, for instance the Hopf fibrations and twistor formalism in ten dimensions. The paramount importance of octonions in the search for unification is based, for instance, in the fact that by extending the division-algebra-valued superalgebras to octonions, in D=11 an octonionic generalized Poincare superalgebra can be constructed, the so-called octonionic M-algebra that describes the octonionic M-theory where the octonionic super-2-brane and the octonionic super-5-brane sectors are shown to be equivalent. Also, there are other vast generalizations and applications of the octonionic formalism such as the classification of quaternionic and octonionic spinors and the pseudo-octonionic formalism.Comment: 13 pages, to appear in Journal of Algebra 302 (1) (2006

Polar analysis of the Dirac equation through dimensions

“…What is interesting, though presumably not surprising, is the fact that this irregular jumping through dimensions of the Clifford algebra, on the one hand, and of the spinor field, on the other hand, renders such a sensitivity to the dimension quite erratic: as is well known, in 4 dimensions the spinor fields have 4 complex components, and thus 8 real components, but only 2 degrees of freedom, whereas in 2 dimensions they have 2 complex components, thus 4 real components, with 3 degrees of freedom. Such results are common knowledge for spinor classification in various dimensions as discussed in [5][6][7][8], where some very general treatments are done which also include the enlargements to arbitrary signatures and up to octonion-valued fields.…”

“…are all real tensors as it is easy to demonstrate. With (7) and these bilinear quantities it is possible to prove the validity of relationships…”

A Note on Laurent Series of the Octonionic Analytic Functions

“…. , e 7 } be the basis of and denote the set W by (1,4,5), (2,4,6), (3,4,7), (2,5,7), (6,1,7), (5,3,6) .…”

Isotopic Liftings of Clifford Algebras and Applications in Elementary Particle Mass Matrices

“…Since ζ is assumed to be always invertible, the product is not automorphic to the product of the original algebras [22]. Given A ∈ A, ζ -applications are defined as…”

“…Also, it leads naturally to remarkable geometric and topological properties, for instance the Hopf fibrations S 3 · · · S 7 → S 4 and S 8 · · · S 15 → S 7 [30][31][32], and twistor formalism in ten dimensions [23,24]. Generalizations of these topics are developed in [22]. We also extend the product ζ to the ζ -isotope (ζ ) A of A, in such a way that for this case we have…”