On the doubling of quadratic algebras

Lars Lindberg¹

¹ Matematiska institutionen Uppsala universitet, Box 480 SE-751 06 Uppsala, Sweden

Colloquium Mathematicum, Tome 100 (2004) no. 1, pp. 119-139.

Résumé

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 $(w,x)(y,z)=(wy-\bar{z}x,x\bar{y}+zw)$. 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.

Détail

DOI : 10.4064/cm100-1-12

Mots-clés : concept doubling which introduced around graves hamilton associates quadratic algebra mathcal field characteristic its double mathcal mathcal mathcal times mathcal multiplication wy overline overline yields endofunctor category quadratic k algebras which faithful full study which respect division property quadratic k algebra preserved under doubling provided whether doubles non isomorphic quadratic division algebras again non isomorphic generalizing theorem dieterich mathbb arbitrary square ordered ground fields prove division property quadratic k algebra dimension smaller equal preserved under doubling generalizing aspect celebrated theorem bott milnor kervaire mathbb arbitrary ground fields characteristic prove division property dimensional doubled quadratic k algebra never preserved under doubling finally contribute solution still problem classifying dimensional real quadratic division algebras extending approach dieterich lindberg proving under mild additional assumption doubles non isomorphic dimensional real quadratic division algebras again non isomorphic