Classification of Some Graded not Necessarily Associative Division Algebras I

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 ∈ 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.

Key Words:

• Division algebras
• Graded algebras
• Non–associative algebras
• Non–commutative algebras
• Twisted group algebras

2010 AMS-MSC:

• 17A35
• 17A01
• 16S35
• 17D99

ACKNOWLEDGEMENTS

This work acquired maturity and some conciseness thanks to profound recommendations of a referee. E.g. one such a recommendation led to Proposition 2(i), another one led to identities (36)–(37).

Notes

Communicated by A. Elduque.