Liftings of dissident maps

Abstract

We study dissident maps η on R m for m ∈ { 3 , 7 } by investigating liftings Φ : R m → R m of the selfbijection η P : P ( R m ) → P ( R m ) , η P [ v ] = ( η ( v ∧ R m ) ) ⊥ induced by η . Our main result (Theorem 2.4) asserts the existence and uniqueness, up to a non-zero scalar multiple, of a lifting Φ whose component functions are homogeneous polynomials of degree d , relatively prime and without non-trivial common zero. We prove that 1 ⩽ d ⩽ m - 2 . We achieve a complete description of all dissident maps of degree one and we solve their isomorphism problem (Theorems 4.8 and 4.13). As a consequence, we achieve a complete description of all real quadratic division algebras of degree one and we solve their isomorphism problem (Theorems 5.1 and 5.3). Moreover we present examples of eight-dimensional real quadratic division algebras of degree 3 and 5 (Proposition 6.3). This extends earlier results of Osborn [Trans. Amer. Math. Soc. 105 (1962) 202–221], Hefendehl [Geometriae Dedicata 9 (1980) 129–152], Hefendehl-Hebeker [Arch. Math. 40 (1983) 50–60], Cuenca Mira et al. [Lin. Alg. Appl. 290 (1999) 1–22], Dieterich [Proc. Amer. Math. Soc. 128 (2000) 3159–3166] and Dieterich and Lindberg [Colloq. Math. 97 (2003) 251–276] on the classification of real quadratic division algebras.