Liftings of dissident maps

We study dissident maps ηη on RmRm for m∈{3,7}m∈{3,7} by investigating liftings Φ:Rm→RmΦ:Rm→Rm of the selfbijection ηP:P(Rm)→P(Rm),ηP[v]=(η(v∧Rm))⊥ induced by ηη. Our main result (Theorem 2.4) asserts the existence and uniqueness, up to a non-zero scalar multiple, of a lifting ΦΦ whose component functions are homogeneous polynomials of degree dd, relatively prime and without non-trivial common zero. We prove that 1⩽d⩽m-21⩽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.