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 R >0 × 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 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 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. Mathematics Subject ClassificationsPresented by Idun Reiten.Dedicated to Claus Michael Ringel on the occasion of his 60th birthday.

Classification, Automorphism Groups and Categorical Structure of the Two-Dimensional Real Division Algebras

“…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 [25], Petersson [33], Hübner and Petersson [29], and Doković and Zhao [23] to the problem of classifying all 2-dimensional real division algebras.…”

“…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 [25], Petersson [33], Hübner and Petersson [29], and Doković and Zhao [23] to the problem of classifying all 2-dimensional real division algebras. …”

“…From Corollary 5.4 and Theorem 5.2 one easily recovers the following alternative classification of all commutative real division algebras according to their automorphism groups, which originally is also from [14]. Let us record one more corollary, describing the structure of the categoryĎ 2 of all 2-dimensional commutative real division algebras whose morphisms are the nonzero algebra morphisms in terms of the matrix groupoid C2 P D3 P, given by the action of C 2 and D 3 respectively on P by conjugation.…”

Some modern developments in the theory of real division algebras

“…In view of Hopf's theorem this, together with the fact that R is the only one-dimensional division algebra, renders a classification in the commutative case. An independent treatment of the commutative division algebras is given in [13]. 2 …”

“…However, due to a technical mistake, the alleged classifying list presented in the article misses some isomorphism classes. See[13] for a detailed account 3. Osborn's claim to have classified all four-dimensional quadratic division algebras over an arbitrary field F of characteristic different from two "modulo the theory of quadratic forms over F", is not accurate.…”