Automorphism groups of real Cayley-Dickson loops

The Cayley-Dickson loop Cn is the multiplicative closure of basic elements of the algebra constructed by n applications of the Cayley-Dickson doubling process (the first few examples of such algebras are real numbers, complex numbers, quaternions, octonions, sedenions). We will discuss properties of the Cayley-Dickson loops, show that these loops are Hamiltonian and describe the structure of their automorphism groups. 1 The Cayley-Dickson doubling process The Cayley-Dickson doubling produces a sequence of power-associative algebras over a field. The dimension of the algebra doubles at each step of the construction. We consider the construction on R, the field of real numbers. Let A0 = R with conjugation a ‡ = a for all a > R. Let An+1 = ˜(a, b) S a, b > An for n > N, where multiplication, addition and conjugation are defined as follows: (a, b)(c, d) = (ac − db, da + bc), (1) (a, b) + (c, d) = (a + c, b + d), (2) (a, b) = (a,−b). (3) Conjugation defines a norm YaY = (aa‡) and the multiplicative inverse for nonzero elements a = a~ YaY. Notice that (a, b)(a, b) = (YaY + YbY ,0) and (a) = a. Dimension of An over R is 2. Definition 1. A nontrivial algebra A over a field is a division algebra if for any nonzero a > A and any b > A there is a unique x > A such that ax = b and a unique y > A such that ya = b. Definition 2. A normed division algebra A is a division algebra over the real or complex numbers which is a normed vector space, with norm YëY satisfying YxyY = YxY YyY for all x, y > A. Theorem 3. (Hurwitz, 1898 [5]) The only normed division algebras over R are A0 = R (real numbers), A1 = C (complex numbers), A2 = H (quaternions) and A3 = O (octonions). 2 Cayley-Dickson loops and their properties We will consider multiplicative structures that arise from the Cayley-Dickson doubling process. Definition 4. A loop is a nonempty set L with binary operation ë such that Department of Mathematics, University of Denver, Denver, CO 80208, USA