Doubled quadratic division algebras

The concept of doubling, introduced around 1840 by Hamilton and Graves, associates with any quadratic algebra A over a field k of characterstic not 2 its double V(A) = A×A, with multiplication (w, x)(y, z) = (wy− z̄x, xȳ + zw). It yields an endofunctor on the category of all quadratic k-algebras which is faithful but not full. We study in which respect the division property of a quadratic k-algebra is preserved under doubling and, provided this is the case, whether the doubles of two non-isomorphic quadratic division algebras again are non-isomorphic. Generalizing a theorem of Dieterich [9] from R to arbitrary square-ordered ground fields k we prove that the division property of a quadratic k-algebra of dimension smaller or equal to 4 is preserved under doubling. Generalizing an aspect of the celebrated (1, 2, 4, 8)-theorem of Bott, Milnor [5] and Kervaire [24] from R to arbitrary ground fields k of characteristic not 2 we prove that the division property of an 8-dimensional doubled quadratic k-algebra never is preserved under doubling. A dissident map on a finite-dimensional Euclidean vector space V is an R-linear map η : V ∧ V → V such that v, w, η(v ∧ w) are linearly independent whenever v, w ∈ V are. The study of dissident maps is one of the major tools when trying to classify the real quadratic division algebras. We will study dissident maps η on Rm by investigating liftings Φ : Rm → Rm of the self-bijection ηP : P(R m) → P(Rm), ηP[v] = (η(v∧R)) induced by η. A major result 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. Finally we combine the concept of doubling with the theory of liftings of dissident maps to contribute to a solution of the still open problem of classifying all 8-dimensional real quadratic division algebras by proving that, under mild additional assumptions, the doubles of two non-isomorphic 4-dimensional real quadratic division algebras again are non-isomorphic. 2000 Mathematics Subject Classification. 15A21, 17A35, 17A45.