VTechWorks is currently (July 30 2016) undergoing indexing to increase the findability of items. During this time some of the search and browse features may behave strangely.

Show simple item record

dc.contributor.advisorGreen, Edwarden_US
dc.contributor.advisorShimozono, Marken_US
dc.contributor.advisorHaskell, Peteren_US
dc.contributor.advisorLetzter, Gailen_US
dc.contributor.authorMcLewin, Kelly Englishen_US
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_US
dc.publisherVirginia Techen_US
dc.rightsThe 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.subjectCayley-Dickson Constructionen_US
dc.subjectExceptional Lie Algebra g2en_US
dc.subjectFano Planeen_US
dc.subjectNormed Division Algebraen_US
dc.titleOctonions and the Exceptional Lie Algebra g_2.en_US

Files in this item


This item appears in the following Collection(s)

Show simple item record