Automorphism Groups of Cayley-Dickson Loops

Jenya Kirshtein

doi:10.4303/jglta/g110901

Abstract

The Cayley-Dickson loop Qn 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, and sedenions).We discuss properties of the Cayley-Dickson loops, show that these loops are Hamiltonian, and describe the structure of their automorphism groups.

Highlights

A loop L is diassociative if every pair of elements of L generates a group in L
We show that the Cayley-Dickson loops are Hamiltonian
Norton [8] formulated a number of theorems characterizing diassociative Hamiltonian loops and showed that the octonion loop is Hamiltonian, at that time he did not study the generalized Cayley-Dickson loops

Summary

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) a, b ∈ An} for n ∈ N, where multiplication, addition, and conjugation are defined as follows:. A normed division algebra A is a division algebra over the real or complex numbers which is a normed vector space, with norm ⋅ satisfying xy = x y for all x, y ∈ A. The only normed division algebras over R are A0 = R (real numbers), A1 = C (complex numbers), A2 = H (quaternions) and A3 = O (octonions). 2010 Mathematics Subject Classification: 20N05, 17D99 Keywords: Cayley-Dickson doubling process, Hamiltonian loop, automorphism group, octonion, sedenion

Cayley-Dickson loops and their properties

Cayley-Dickson loops are Hamiltonian

Automorphism groups of the Cayley-Dickson loops

The size of D is