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.