| dc.description | Introducing products between multivectors of Cl-0,Cl-7 (the Clifford algebra over the metric vector space R-0,R-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 S-7, and the XY-product. This generalization is accomplished in the u- and (u, v)-products, where u, v is an element of Cl-0,Cl-7 are fixed, but arbitrary. Moreover, we extend these original products in order to encompass the most general-non-associative-products (R circle plus R-0,R-7) x Cl-0,Cl-7 -> R circle plus R-0,R-7, Cl-0,Cl-7 x (R circle plus R-0,R-7) -> R circle plus R-0,R-7 and Cl-0,Cl-7 x Cl-0,Cl-7 -> R circle plus R-0,R-7. We also present the formalism necessary to construct Clifford algebra-parametrized octonions, which provides the structure to present the O-1,O-u algebra. Finally we introduce a method to construct O-algebras endowed with the (u, v)-product from O-algebras endowed with the u-product. These algebras are called O-like algebras and their octonionic units are parametrized by arbitrary Clifford multivectors. When u is restricted to the underlying paravector space R circle plus R-0,R-7 -> Cl-0,Cl-7 of the octonion algebra O, these algebras are shown to be isomorphic. 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. (c) 2006 Published by Elsevier Inc. | |
