Clifford algebra-parametrized octonions and generalizations

Introducing products between multivectors of (the Clifford algebra over the metric vector space ) 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 S7, and the XY-product. This generalization is accomplished in the u- and (u,v)-products, where are fixed, but arbitrary. Moreover, we extend these original products in order to encompass the most general—non-associative—products , and . We also present the formalism necessary to construct Clifford algebra-parametrized octonions, which provides the structure to present the algebra. Finally we introduce a method to construct -algebras endowed with the (u,v)-product from -algebras endowed with the u-product. These algebras are called -like algebras and their octonionic units are parametrized by arbitrary Clifford multivectors. When u is restricted to the underlying paravector space of the octonion algebra , 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.