Classification, Automorphism Groups and Categorical Structure of the Two-Dimensional Real Division Algebras
نویسنده
چکیده
The category of all 2-dimensional real division algebras is shown to split into four full subcategories each of which is given by the natural action of a Coxeter group of type A1 or A2 on the set of all pairs of ellipses in R which are centred in the origin and have reciprocal axis lengths. Cross-sections for the orbit sets of these group actions are being determined. They yield a classification of all 2-dimensional real division algebras. Moreover all morphisms between the objects in this classifying list are described, and thus an explicit and geometric picture of the category of all 2-dimensional real division algebras is obtained. This elementary and self-contained exposition extends Darpö and Dieterich’s recent description [14] of the category of all 2-dimensional commutative real division algebras, which in turn is based on Benkart, Britten and Osborn’s investigation [4] of the isotopes of C. It also supplements earlier contributions of Althoen and Kugler [2], Burdujan [9], Gottschling [24], and Hübner and Petersson [28] to the problem of classifying all 2-dimensional real division algebras. Mathematics Subject Classification 2000: 15A21, 15A22, 15A23, 17A35, 17A36.
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