Real Commutative Division Algebras

Abstract

The category of all two-dimensional real commutative division algebras is shown to split into two full subcategories. One of them is equivalent to the category of the natural action of the cyclic group of order 2 on the open right half plane $\mathbb{R}_{{ > 0}} \times \mathbb{R}$. The other one is equivalent to the category of the natural action of the dihedral group of order 6 on the set of all ellipses in $\mathbb{R}^{2}$ which are centered at the origin and have reciprocal axis lengths. Cross-sections for the orbit sets of these group actions are easily described. Together with $\mathbb{R}$ they classify all real commutative division algebras up to isomorphism. Moreover we describe all morphisms between the objects in this classifying set, thus obtaining a complete picture of the category of all real commutative division algebras, up to equivalence. This supplements earlier contributions of Kantor and Solodovnikov, Hypercomplex Numbers: An Elementary Introduction to Algebras, Nauka, Moscow, 1973; Benkart et al., Hadronic J., 4: 497–529, 1981; and Althoen and Kugler, Amer. Math. Monthly, 90: 625–635, 1983, who achieved partial results on the classification of the real commutative division algebras.