On the doubling of quadratic algebras

• Autorzy: Lars Lindberg
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 15A21: Canonical forms, reductions, classification
• 17A45: Quadratic algebras (but not quadratic Jordan algebras)
• 17A35: Division algebras

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2004
• Tom: 100
• Numer: 1
• Strony: 119-139

Daty

wydano

2004

Twórcy

autor

Lars Lindberg

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

Identyfikatory

DOI

10.4064/cm100-1-12