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Cayley-Dickson Loops, Jenya Kirshtein

Electronic Theses and Dissertations

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 Q_n 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 Q_n. We establish or recall elementary facts about Q_n, e.g., inverses, …