Dissident maps on the seven-dimensional Euclidean space

Ernst Dieterich¹ ; Lars Lindberg¹

¹ Matematiska institutionen Uppsala universitet, Box 480 SE-751 06 Uppsala, Sweden

Colloquium Mathematicum, Tome 97 (2003) no. 2, pp. 251-276.

Résumé

Our article contributes to the classification of dissident maps on ${\mathbb R}^7$, which in turn contributes to the classification of 8-dimensional real division algebras. We study two large classes of dissident maps on ${\mathbb R}^7$. The first class is formed by all composed dissident maps, obtained from a vector product on ${\mathbb R}^7$ 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 ${\mathbb R}^7$.

Détail

DOI : 10.4064/cm97-2-10

Keywords: article contributes classification dissident maps mathbb which turn contributes classification dimensional real division algebras study large classes dissident maps mathbb first class formed composed dissident maps obtained vector product mathbb composition definite endomorphism second class formed doubled dissident maps obtained purely imaginary parts structures those dimensional real quadratic division algebras which arise dimensional real quadratic division algebra doubling each these classes exhibit complete redundant classification given parameter family composed dissident maps parameter family doubled dissident maps respectively intersection these classes forms isoclass dissident maps only namely isoclass consisting vector products mathbb