Quadratic Division Algebras Revisited ( Remarks On

In his remarkable article “Quadratic division algebras” (Trans. Amer. Math. Soc. 105 (1962), 202–221), J. M. Osborn claims to solve ‘the problem of determining all quadratic division algebras of order 4 over an arbitrary field F of characteristic not two . . . modulo the theory of quadratic forms over F ’ (cf. p. 206). While we shall explain in which respect he has not achieved this goal, we shall on the other hand complete Osborn’s basic results (by a reasoning which is finer than his) to derive in the real ground field case a classification of all 4-dimensional quadratic division algebras and the construction of a 49-parameter family of pairwise nonisomorphic 8-dimensional quadratic division algebras. To make these points clear, we begin by reformulating Osborn’s fundamental observations on quadratic algebras in categorical terms. 1. First equivalence for anisotropic quadratic algebras Let k be a field of characteristic not two. An algebra is a k-vectorspaceA endowed with a k-bilinear multiplication A × A → A, (x, y) 7→ xy. A nonzero algebra A is called quadratic in case an identity element 1 ∈ A exists and each x ∈ A satisfies an equation x = αx + β1 with α, β ∈ k. Given any quadratic algebra A, Frobenius’ lemma states that the set V = {v ∈ A | v ∈ k1} \ (k1 \ {0}) of purely imaginary elements of A forms a linear subspace of A which is supplementary to k1 (cf. [9],[3],[11]). Accordingly, each x ∈ A has unique decomposition x = λ(x)1 + ι(x), with λ(x) ∈ k and ι(x) ∈ V . The linear form λ : A → k gives rise to the bilinear form ( ) : V × V → k, (x, y) = −λ(xy), and the projection ι : A→ V gives rise to the bilinear multiplication ◦ : V × V → V, v ◦ w = ι(vw) which by Frobenius’ lemma is anticommutative. Let us introduce the category A of all quadratic k-algebras, and the category A of all anticommutative k-algebras endowed with a bilinear form. Morphisms in A are algebra morphisms respecting identity elements, while morphisms in A are algebra morphisms respecting bilinear forms. In these terms, Frobenius’ lemma gives rise to the map of object classes Φ : Ob(A)→ Ob(A), Φ(A) = (V, ◦, ( )). Received by the editors December 8, 1998 and, in revised form, January 4, 1999. 2000 Mathematics Subject Classification. Primary 17A35, 17A45, 57S25. c ©2000 American Mathematical Society