Classification of some graded not necessarily associative division algebras I

Details Classification of some graded not necessarily associative division algebras I Classification of some graded not necessarily associative division algebras I

2010-07-09

arxiv.org/abs/1007.3730v2

Mathematics

Rings and Algebras

Scientific paper

We study not necessarily associative (NNA) division algebras over the reals. We classify 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 in $\{1,-1\}$. That is, the algebra has a twisted product, and it is a twisted group algebra. We classify those graded by a group $G$ of order $|G|\leq 4$. We will find the complex, and quaternion algebras, but also a remarkable set of novel non--associative division algebras.

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