Generalized non-associative structures on the 7-sphere

TL;DR

This paper introduces a broad class of non-associative products on the 7-sphere using Clifford and exterior bundles, extending previous algebraic frameworks and exploring their properties and potential physical applications.

Contribution

It develops a generalized framework for non-associative structures on S7, incorporating Clifford algebra, exterior bundles, and octonions, with new properties and involutions.

Findings

New non-associative products on S7 are constructed.

Generalizations include involutions and shear transformations.

Potential applications in physics are suggested.

Abstract

In this paper we provide a more general class of non-associative products using the exterior and Clifford bundles on the 7-sphere. Some additional properties encompass previous formalisms in the Clifford algebra context, and wider classes of non-associative structures on the 7-sphere 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 Cl(0,7) elements into octonions. We assert new properties inherited from the non-associative structure introduced in the whole Clifford bundle on S7, which naturally induce involutions on the Clifford bundle and provide immediate generalizations concerning well-established formal results and potential applications in physics.