Composition Algebras and Their Automorphisms

Abstract

The principal objective of the present paper is the study of the automorphisms and groups of automorphisms of composition algebras, that is, the algebras arising from quadratic forms which permit composition. These algebras are mainly quaternion algebras and Cayley algebras. The problem of determining the quadratic forms which permit composition (Huryitz’s problem) has been treated by many authors (2). In spite of this, there does not appear in any one place a complete solution of this problem in its most general form — which amounts to the determination of the algebras for an arbitrary field and not just to the determination of the possible dimensionalities. We give such a solution here for the case of characteristic not two. Aside from its intrinsic interest and applications to other fields (for example Jordan algebras, absolute valued algebras) we have still another reason for treating the Hurvitz problem again, namely: The analysis of the composition algebras is essential for our study of their automorphisms.