Octonions and the Exceptional Lie Algebra g_2

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dc.contributor.authorMcLewin, Kelly Englishen
dc.contributor.committeechairLetzter, Gailen
dc.contributor.committeememberGreen, Edward L.en
dc.contributor.committeememberShimozono, Mark M.en
dc.contributor.committeememberHaskell, Peter E.en
dc.contributor.departmentMathematicsen
dc.date.accessioned2011-08-06T16:01:23Zen
dc.date.adate2004-04-28en
dc.date.available2011-08-06T16:01:23Zen
dc.date.issued2004-04-01en
dc.date.rdate2004-04-28en
dc.date.sdate2004-04-23en
dc.description.abstractWe first introduce the octonions as an eight dimensional vector space over a field of characteristic zero with a multiplication defined using a table. We also show that the multiplication rules for octonions can be derived from a special graph with seven vertices call the Fano Plane. Next we explain the Cayley-Dickson construction, which exhibits the octonions as the set of ordered pairs of quaternions. This approach parallels the realization of the complex numbers as ordered pairs of real numbers. The rest of the thesis is devoted to following a paper by N. Jacobson written in 1939 entitled "Cayley Numbers and Normal Simple Lie Algebras of Type G". We prove that the algebra of derivations on the octonions is a Lie algebra of type G_2. The proof proceeds by showing the set of derivations on the octonions is a Lie algebra, has dimension fourteen, and is semisimple. Next, we complexify the algebra of derivations on the octonions and show the complexification is simple. This suffices to show the complexification of the algebra of derivations is isomorphic to g_2 since g_2 is the only semisimple complex Lie algebra of dimension fourteen. Finally, we conclude the algebra of derivations on the octonions is a simple Lie algebra of type G_2.en
dc.description.degreeMaster of Scienceen
dc.format.mediumETDen
dc.identifier.otheretd-04232004-131642en
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-04232004-131642en
dc.identifier.urihttp://hdl.handle.net/10919/9878en
dc.publisherVirginia Techen
dc.relation.haspartthesis.pdfen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectCayley-Dickson Constructionen
dc.subjectOctonionen
dc.subjectExceptional Lie Algebra g2en
dc.subjectFano Planeen
dc.subjectDerivationen
dc.subjectNormed Division Algebraen
dc.titleOctonions and the Exceptional Lie Algebra g_2en
dc.typeThesisen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.levelmastersen
thesis.degree.nameMaster of Scienceen

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