Date of Award

1-1-2012

Document Type

Dissertation

Degree Name

Ph.D.

Organizational Unit

Mathematics

First Advisor

Petr Vojtechovsky, Ph.D.

Second Advisor

Michael Kinyon

Third Advisor

Richard Ball

Fourth Advisor

Nikolaos Galatos

Fifth Advisor

Richard Green

Sixth Advisor

J. Michael Daniels

Keywords

Cayley-Dickson doubling process, Loop theory, Multiplication group, Nonassociative, Octonion, Sedenion

Abstract

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 begin by describing basic properties of the Cayley–Dickson loops Qn. We establish or recall elementary facts about Qn, e.g., inverses, conjugates, orders of elements, and diassociativity. We then discuss some important subloops of Qn, for instance, associator subloop, derived subloop, nuclei, center, and show that Qn are Hamiltonian. We study the structure of the automorphism groups of Qn. 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 O16, or the quasioctonion loop O16. This helps to establish that starting at the sedenion loop, the group Aut (Qn) is isomorphic to Aut (O16) x (Z2) n−3 .

Next we study two notions that are of interest in loop theory, the inner mapping group Inn(Qn) and the multiplication group Mlt(Qn). We prove that Inn(Qn) is an elementary abelian 2-group of order 22 n−2 , moreover, every f > Inn(Q) is a product of disjoint transpositions of the form (x,−x). This implies that nonassociative Cayley–Dickson loops are not automorphic. The elements of Mlt(Qn) are even permutations and have order 1, 2 or 4. We show that Mlt(Qn) is a semidirect product of Inn(Qn) x Z2 and an elementary abelian 2-group K, and construct an isomorphic copy of Mlt(Qn) as an external semidirect product of two abstract elementary abelian 2-groups. The groups Innl(Qn) and Innr(Qn) are proved to be equal, elementary abelian 2-groups of order 22 n−1−1 . We also establish that Mltl(Qn) is a semidirect product of Innl(Qn) x Z2 and K, and that Mltl(Qn) and Mltr(Qn) are isomorphic.

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.

Publication Statement

Copyright is held by the author. User is responsible for all copyright compliance.

Rights Holder

Jenya Kirshtein

Provenance

Received from ProQuest

File Format

application/pdf

Language

en

File Size

119 p.

Discipline

Mathematics

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Mathematics Commons

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