Dissident maps on the seven-dimensional Euclidean space

Abstract

Our article contributes to the classification of dissident maps on ℝ7, which in turn contributes to the classification of 8-dimensional real division algebras. We study two large classes of dissident maps on ℝ7. The first class is formed by all composed dissident maps, obtained from a vector product on ℝ7 by composition with a definite endomorphism. The second class is formed by all doubled dissident maps, obtained as the purely imaginary parts of the structures of those 8-dimensional real quadratic division algebras which arise from a 4-dimensional real quadratic division algebra by doubling. For each of these two classes we exhibit a complete (but redundant) classification, given by a 49-parameter family of composed dissident maps and a 9-parameter family of doubled dissident maps respectively. The intersection of these two classes forms one isoclass of dissident maps only, namely the isoclass consisting of all vector products on ℝ7.