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

Abstract

In this paper we discuss generalized properties of non-associativity in Clifford bundles on the 7-sphere S 7 . Novel and prominent properties inherited from the non-associative structure of the Clifford bundle on S 7 are demonstrated. They naturally lead to general transformations of the spinor fields on S 7 and have dramatic consequences for the associated Kač-Moody current algebras. All additional properties concerning the non-associative structure in the Clifford bundle on S 7 are considered. We further discuss and explore their applications.Mathematics Subject Classification (2010). 15A66, 11E88, 55R25.

Additional fermionic fields onto parallelizable 7-spheres

“…, is an isomorphism, for all a ∈ R and v ∈ R 0,7 [29]. The reciprocal statement is up to now a conjecture.…”

“…According to the Radon-Hurwitz theorem in the Appendix B, for k = 7 − r 7 = 4, one aims the set {e I 1 , e I 2 , e I 3 , e I 4 } ⊂ Cℓ 0,7 that commute and squares the identity [28]. Identifying, for example [29] is a primitive one. Hence, a spinor ψ ∈ S 7 has its algebraic version as the elementψf of the left ideal Cℓ 0,7 f , for some multivectorψ ∈ Cℓ 0,7 .…”

“…֒→ sec Cℓ 0,7 , where {u p } k p=1 ⊂ sec T R 0,7 and A ∈ sec(OS 7 ), the products • and • are defined (and extended by linearity) by [19,29] • : sec(OS 7 ) × sec Λ k (R 0,7 ) → sec(OS 7 )…”

“…According to the Radon-Hurwitz theorem in the Appendix B, for k = 7 − r 7 = 4, one aims the set {e I 1 , e I 2 , e I 3 , e I 4 } ⊂ Cℓ 0,7 that commute and squares the identity[28]. Identifying, for example[29] e I 1 = e 1 e 2 e 3 , e I ∈ Cℓ 0,7…”