Octonions and the Exceptional Lie Algebra g_2.

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dc.contributor.advisor Green, Edward en_US
dc.contributor.advisor Shimozono, Mark en_US
dc.contributor.advisor Haskell, Peter en_US
dc.contributor.advisor Letzter, Gail en_US
dc.contributor.author McLewin, Kelly English en_US
dc.date.accessioned 2011-08-06T16:01:23Z
dc.date.available 2011-08-06T16:01:23Z
dc.date.issued 2004-04-01 en_US
dc.identifier.other etd-04232004-131642 en_US
dc.identifier.uri http://hdl.handle.net/10919/9878
dc.description.abstract We 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_US
dc.format.medium ETD en_US
dc.publisher Virginia Tech en_US
dc.relation.haspart thesis.pdf en_US
dc.rights The authors of the theses and dissertations are the copyright owners. Virginia Tech's Digital Library and Archives has their permission to store and provide access to these works. en_US
dc.source.uri http://scholar.lib.vt.edu/theses/available/etd-04232004-131642 en_US
dc.subject Cayley-Dickson Construction en_US
dc.subject Octonion en_US
dc.subject Exceptional Lie Algebra g2 en_US
dc.subject Fano Plane en_US
dc.subject Derivation en_US
dc.subject Normed Division Algebra en_US
dc.title Octonions and the Exceptional Lie Algebra g_2. en_US
dc.type Thesis en_US
dc.contributor.department Mathematics en_US
dc.description.degree MS en_US

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