Automorphism groups of Cayley-Dickson loops

TL;DR

This paper investigates the properties of Cayley-Dickson loops, demonstrating they are Hamiltonian and detailing the structure of their automorphism groups, thus advancing understanding of algebraic symmetries in these complex systems.

Contribution

It provides a detailed analysis of the automorphism groups of Cayley-Dickson loops, revealing their structure and Hamiltonian nature, which was previously not fully understood.

Findings

Cayley-Dickson loops are Hamiltonian.

Automorphism groups of Cayley-Dickson loops are characterized.

Properties of these loops relate to algebraic symmetries.

Abstract

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