On the doubling of quadratic algebras
Volume 100 / 2004
Abstract
The concept of doubling, which was introduced around 1840 by Graves and Hamilton, associates with any quadratic algebra ${\mathcal A}$ over a field $k$ of characteristic not 2 its double ${\mathcal V}({\mathcal A})={\mathcal A}\times {\mathcal A}$ with multiplication $(w,x)(y,z)=(wy-\overline {z}x,x\overline {y}+zw)$. This 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 are again non-isomorphic.
Generalizing a theorem of Dieterich [9] from ${\mathbb R}$ to arbitrary square-ordered ground fields $k$ we prove that the division property of a quadratic $k$-algebra of dimension smaller than or equal to 4 is preserved under doubling. Generalizing an aspect of the celebrated (1,2,4,8)-theorem of Bott, Milnor [4] and Kervaire [21] from ${\mathbb R}$ to arbitrary ground fields $k$ of characteristic not 2 we prove that the division property of an $8$-dimensional doubled quadratic $k$-algebra is never preserved under doubling. Finally, we contribute to a solution of the still open problem of classifying all $8$-dimensional real quadratic division algebras by extending an approach of Dieterich and Lindberg [12] and proving that, under a mild additional assumption, the doubles of two non-isomorphic 4-dimensional real quadratic division algebras are again non-isomorphic.
