Cayley-Dickson algebras

This thesis studies the various effects of the nonassociativity of the Cayley-Dickson algebras At, t>3, especially on the structure of their automorphism groups. Beginning with the problem of composition algebra structures on euclidean spaces, we shall explain the origin of the Cayley-Dickson algebras, and give a self-contained exposition on some important results on such algebras. These algebras being nonassociative, we focus on the study of the associators of the form (u,w,v) = (uw)v - u(wv). The first main result, that if u and v are elements in a Cayley-Dickson algebra for which (u, w, v) = 0 for all w, then u and v generate a 2-dimensional subalgebra isomorphic to C, was conjectured by P. Yiu, and proved by P. Eakin and A. Sathaye. We shall simplify the proof given by these latter authors. This is then used to give a simple proof of R. D. Schafer's theorem on derivations of Cayley-Dickson algebras, and following also Eakin and Sathaye, a proof of the conjecture by R. B. Brown on the structure of the automorphism groups of these algebras. Two simple proofs are presented for the beautiful characterization by H. Brandt that in the Cayley algebra A3 = K, conjugation by a unit element a is an automorphism if and only if a is a 6th root of unity. We shall present a geometric proof by M. A. Zorn and a purely algebraic one. The zero divisors of the Cayley-Dickson algebra A4 are also analyzed in detail.