Generalized non-associative structures on the 7-sphere

Abstract

In this paper we provide a more general class of non-associative products using the exterior and Clifford bundles on the 7-sphere S 7 . Some additional properties encompass the previous formalisms presented in [1,2] in the Clifford algebra context, and wider classes of non-associative structures on S 7 are investigated, evinced by the directional non-associative products and the mixed composition of generalized non-associative products between Clifford algebra multivectors. These non-associative products are further generalized by considering the non-associative shear of arbitrary Clifford bundle Cℓ0,7 elements into octonions. We assert new properties inherited from the non-associative structure introduced in the whole Clifford bundle on S 7 , which naturally induce involutions on the Clifford bundle and provide immediate generalizations concerning well-established formal results in, e.g., [3,4,5,6,7] and potential applications in physics as pioneered by, e. g., [2,3,4,5,6,7,8,9,10,11,12,13,14].

Non-Associativity in the Clifford Bundle on the Parallelizable Torsion 7-Sphere

“…The Clifford product between w ∈ V and ψ ∈ V is given by wψ = w ∧ ψ + w ⌋ ψ. The Grassmann algebra ( V, g) endowed with the Clifford product is denoted by Cℓ(V, g) or Cℓ p,q , the Clifford algebra associated with V ≃ R p,q , p + q = n. The octonionic algebra via the Clifford algebras is briefly reviewed in [1][2][3]. The octonionic algebra O can be defined as the paravector space R ⊕ R 0,7 endowed with the product •.…”

“…holds in a context where any similar generalization related to (3.1) can be constructed in the non-associative formalism induced by the u-product for u ∈ sec T x S 7 , the exterior bundle on S 7 . In a straightforward example we showed in [2] that, in general,…”

“…It was seen in [2] that (e 57 − e 31 ) ⊙ e 123 = e 7 + e 2 , while (e 57 − e 31 ) ⊙ e 123 = −e 7 + e 2 .…”

“…The main goal of this paper is provide a general class of non-associative structures on S 7 in the Clifford algebra formulation of generalized octonionic products. After briefly revisiting previous results [1,2], an octonionic product is defined in the Clifford algebra Cℓ 0,7 [3] which is closely related to the S 7 sphere [8]. Using this formalism, certain new identities are derived in a generalized octonionic algebra.…”

“…The results from [1,2] are generalized along with a definition of nonequivalent non-associative products which are introduced in order to discuss the Clifford bundle on S 7 [1]. The formalism presented in [4,6] shows that the octonionic product can be deformed in order to include a parallelizable torsion on S 7 .…”