Classification of some graded not necessarily associative division algebras I (1007.3730v4)

Abstract: We study not necessarily associative (NNA) division algebras over the reals. We classify in this paper series those that admit a grading over a finite group $G$, and have a basis ${v_g|g\in G}$ as a real vector space, and the product of these basis elements respects the grading and includes a scalar structure constant with values only in ${1,-1}$. We classify here those graded by an abelian group $G$ of order $|G|\leq 8$ with $G$ non--isomorphic to $\z/8\z$. We will find the complex, quaternion, and octonion algebras, but also a remarkable set of novel non--associative division algebras.