CLASSIFICATION OF SOME GRADED NOT NECESSARILY ASSOCIATIVE DIVISION ALGEBRAS I

摘要

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_gg ∈ 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 ≤ 8 with G non–isomorphic to Z/8Z. We will find the complex, quaternion, and octonion algebras, but also a remarkable set of novel non–associative division algebras.