CLASSIFICATION, AUTOMORPHISM GROUPS AND CATEGORICAL STRUCTURE OF THE TWO-DIMENSIONAL REAL DIVISION ALGEBRAS

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 A(1) or A(2) on the set of all pairs of ellipses in R-2, 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 Darp " o 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 [25], Petersson [33], Hubner and Petersson [29], and Dokovi ' c and Zhao [23] to the problem of classifying all 2-dimensional real division algebras.