On locally complex algebras and low-dimensional Cayley-Dickson algebras

Details On locally complex algebras and low-dimensional Cayley-Dickson algebras On locally complex algebras and low-dimensional Cayley-Dickson algebras

2010-10-11

arxiv.org/abs/1010.2156v2

Mathematics

Rings and Algebras

Some changes suggested by the referee, accepted for publication in Journal of Algebra

Scientific paper

The paper begins with short proofs of classical theorems by Frobenius and (resp.) Zorn on associative and (resp.) alternative real division algebras. These theorems characterize the first three (resp. four) Cayley-Dickson algebras. Then we introduce and study the class of real unital nonassociative algebras in which the subalgebra generated by any nonscalar element is isomorphic to C. We call them locally complex algebras. In particular, we describe all such algebras that have dimension at most 4. Our main motivation, however, for introducing locally complex algebras is that this concept makes it possible for us to extend Frobenius' and Zorn's theorems in a way that it also involves the fifth Cayley-Dickson algebra, the sedenions.

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