The concept of doubling, which was introduced around 1840 by Graves and Hamilton, associates with any quadratic algebra 𝓐 over a field k of characteristic not 2 its double 𝓥(𝓐 ) = 𝓐 × 𝓐 with multiplication (w,x)(y,z) = (wy - z̅x,xy̅ + 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 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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm100-1-12,
author = {Lars Lindberg},
title = {On the doubling of quadratic algebras},
journal = {Colloquium Mathematicae},
volume = {100},
year = {2004},
pages = {119-139},
zbl = {1061.17003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm100-1-12}
}
Lars Lindberg. On the doubling of quadratic algebras. Colloquium Mathematicae, Tome 100 (2004) pp. 119-139. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm100-1-12/