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On the doubling of quadratic algebras

100%

Lindberg L.

Colloquium Mathematicum

2004

tom 100

nr 1

119-139

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.

Dissident maps on the seven-dimensional Euclidean space

63%

Dieterich E., Lindberg L.

Colloquium Mathematicum

2003

tom 97

nr 2

251-276

Our article contributes to the classification of dissident maps on ℝ ⁷, which in turn contributes to the classification of 8-dimensional real division algebras. We study two large classes of dissident maps on ℝ ⁷. The first class is formed by all composed dissident maps, obtained from a vector product on ℝ ⁷ by composition with a definite endomorphism. The second class is formed by all doubled dissident maps, obtained as the purely imaginary parts of the structures of those 8-dimensional real quadratic division algebras which arise from a 4-dimensional real quadratic division algebra by doubling. For each of these two classes we exhibit a complete (but redundant) classification, given by a 49-parameter family of composed dissident maps and a 9-parameter family of doubled dissident maps respectively. The intersection of these two classes forms one isoclass of dissident maps only, namely the isoclass consisting of all vector products on ℝ ⁷.

Ograniczanie wyników

2 Colloquium Mathematicum

2 Lindberg L.

1 Dieterich E.

1 2004

1 2003

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On the doubling of quadratic algebras

Dissident maps on the seven-dimensional Euclidean space