Duplication methods for embeddings of real division algebras

Abstract

We introduce two groups of duplication processes that extend the well known Cayley–Dickson process. The first one allows to embed every [Formula: see text]-dimensional (4D) real unital algebra [Formula: see text] into an 8D real unital algebra denoted by [Formula: see text] We also find the conditions on [Formula: see text] under which [Formula: see text] is a division algebra. This covers the most classes of known [Formula: see text]D real division algebras. The second process allows us to embed particular classes of [Formula: see text]D RDAs into [Formula: see text]D RDAs. Besides, both duplication processes give an infinite family of non-isomorphic [Formula: see text]D real division algebras whose derivation algebras contain [Formula: see text].