On Finite-Dimensional Absolute Valued Algebras

This is a study of morphisms in the category of finite dimensional absolute valued algebras, whose codomains have dimension four. We begin by citing and transferring a classification of an equivalent category. Thereafter, we give a complete description of morphisms from one-dimensional algebras, partly via solutions of real polynomials, and a complete, explicit description of morphisms from two-dimensional algebras. We then give an account of the reducibility of the morphisms, and for the morphisms from two-dimensional algebras we describe the orbits under the actions of the automorphism groups involved. Parts of these descriptions rely on a suitable choice of a cross-section of four-dimensional absolute valued algebras, and we thus end by providing an explicit means of transferring these results to algebras outside this crosssection. 1. Definitions and background An algebra A = (A, ·) over a field k is a vector space A over k equipped with a k-bilinear multiplication A×A→ A, (x, y) 7→ xy = x · y. Neither associativity nor commutativity is in general assumed. A is called unital if it contains an element neutral under multiplication; in that case, such an element is unique, and will be denoted by 1. If A is non-zero, and if for each a ∈ A \ {0}, the maps La : A → A, x 7→ ax and Ra : A → A, x 7→ xa are bijective, A is called a division algebra. This implies that A has no zero divisors and, if the dimension of A is finite, it is equivalent to having no zero divisors. An algebra A is called absolute valued if the vector space is real and equipped with a norm ‖ · ‖ such that ‖xy‖ = ‖x‖‖y‖ for all x, y ∈ A. By [1] the norm in a finite dimensional absolute valued algebra is uniquely determined by the algebra multiplication if the algebra has finite dimension. The multiplicativity of the norm implies that an absolute valued algebra has no zero divisors and hence, if it is finite dimensional, that it is a division algebra. The class of all finite dimensional absolute valued algebras forms a category A, in which the morphisms are the nonzero algebra homomorphisms. Thus A is a full subcategory of the category D(R) of finite dimensional real division algebras. It is known that morphisms in A respect the norm, and are hence injective. (Injectivity in fact holds for all morphisms in D(R).) 2010 Mathematics Subject Classification. 17A35; 17A80.