Date of Award
Cayley-Dickson doubling process, loop theory, multiplication group, nonassociative, octonion, sedenion
In this dissertation we study the Cayley-Dickson loops, multiplicative structures arising from the standard Cayley-Dickson doubling process. More precisely, the Cayley-Dickson loop Qn is the multiplicative closure of basic elements of the algebra constructed by n applications of the doubling process (the first few examples of such algebras are real numbers, complex numbers, quaternions, octonions, sedenions). Starting at the octonions, Cayley-Dickson algebras and loops become nonassociative, which presents a significant challenge in their study.
We describe basic properties of the Cayley-Dickson loops Qn, e.g., inverses, conjugates, orders of elements, and diassociativity. We discuss some important subloops of Qn, for instance, associator subloop, derived subloop, nuclei, center, and show that Qn are
Hamiltonian. We show that all subloops of Qn of order 16 fall into two isomorphism classes, in particular, any such subloop is either isomorphic to the octonion loop, or the quasioctonion loop. We discuss automorphism groups, inner mapping groups, and multiplication groups of the Cayley-Dickson loops, and describe the progress made on the study of their subloop structure. We also provide incidence tetrahedra for the sedenion loop and other subloops of order 32, generalizing the idea of the octonion multiplication Fano plane.
Kirshtein, Jenya, "Cayley-Dickson Loops" (2012). Electronic Theses and Dissertations. 849.
Recieved from ProQuest