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CayleyDickson algebras
 Date Issued:
 1993
 Summary:
 This thesis studies the various effects of the nonassociativity of the CayleyDickson algebras At, t>3, especially on the structure of their automorphism groups. Beginning with the problem of composition algebra structures on euclidean spaces, we shall explain the origin of the CayleyDickson algebras, and give a selfcontained exposition on some important results on such algebras. These algebras being nonassociative, we focus on the study of the associators of the form (u,w,v) = (uw)v  u(wv). The first main result, that if u and v are elements in a CayleyDickson algebra for which (u, w, v) = 0 for all w, then u and v generate a 2dimensional subalgebra isomorphic to C, was conjectured by P. Yiu, and proved by P. Eakin and A. Sathaye. We shall simplify the proof given by these latter authors. This is then used to give a simple proof of R. D. Schafer's theorem on derivations of CayleyDickson algebras, and following also Eakin and Sathaye, a proof of the conjecture by R. B. Brown on the structure of the automorphism groups of these algebras. Two simple proofs are presented for the beautiful characterization by H. Brandt that in the Cayley algebra A3 = K, conjugation by a unit element a is an automorphism if and only if a is a 6th root of unity. We shall present a geometric proof by M. A. Zorn and a purely algebraic one. The zero divisors of the CayleyDickson algebra A4 are also analyzed in detail.
Title:  The CayleyDickson algebras. 
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Name(s): 
Khalil, Saidah Hasan Florida Atlantic University, Degree grantor Yiu, Paul Y., Thesis advisor Charles E. Schmidt College of Science Department of Mathematical Sciences 

Type of Resource:  text  
Genre:  Electronic Thesis Or Dissertation  
Issuance:  monographic  
Date Issued:  1993  
Publisher:  Florida Atlantic University  
Place of Publication:  Boca Raton, FL  
Physical Form:  application/pdf  
Extent:  65 p.  
Language(s):  English  
Summary:  This thesis studies the various effects of the nonassociativity of the CayleyDickson algebras At, t>3, especially on the structure of their automorphism groups. Beginning with the problem of composition algebra structures on euclidean spaces, we shall explain the origin of the CayleyDickson algebras, and give a selfcontained exposition on some important results on such algebras. These algebras being nonassociative, we focus on the study of the associators of the form (u,w,v) = (uw)v  u(wv). The first main result, that if u and v are elements in a CayleyDickson algebra for which (u, w, v) = 0 for all w, then u and v generate a 2dimensional subalgebra isomorphic to C, was conjectured by P. Yiu, and proved by P. Eakin and A. Sathaye. We shall simplify the proof given by these latter authors. This is then used to give a simple proof of R. D. Schafer's theorem on derivations of CayleyDickson algebras, and following also Eakin and Sathaye, a proof of the conjecture by R. B. Brown on the structure of the automorphism groups of these algebras. Two simple proofs are presented for the beautiful characterization by H. Brandt that in the Cayley algebra A3 = K, conjugation by a unit element a is an automorphism if and only if a is a 6th root of unity. We shall present a geometric proof by M. A. Zorn and a purely algebraic one. The zero divisors of the CayleyDickson algebra A4 are also analyzed in detail.  
Identifier:  14993 (digitool), FADT14993 (IID), fau:11772 (fedora)  
Degree granted:  Thesis (M.S.)Florida Atlantic University, 1993.  
Collection:  FAU Electronic Theses and Dissertations Collection  
Note(s):  Charles E. Schmidt College of Science  
Subject(s):  Cayley algebras  
Held by:  Florida Atlantic University Libraries  
Persistent Link to This Record:  http://purl.flvc.org/fcla/dt/14993  
Sublocation:  Digital Library  
Use and Reproduction:  Copyright © is held by the author with permission granted to Florida Atlantic University to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.  
Use and Reproduction:  http://rightsstatements.org/vocab/InC/1.0/  
Host Institution:  FAU  
Is Part of Series:  Florida Atlantic University Digital Library Collections. 