Type of Document	Master's Thesis
Author	McLewin, Kelly English
URN	etd-04232004-131642
Title	Octonions and the Exceptional Lie Algebra g_2.
Degree	Master of Science
Department	Mathematics
Advisory Committee	Advisor Name Title Letzter, Gail Committee Chair Green, Edward L. Committee Member Haskell, Peter E. Committee Member Shimozono, Mark M. Committee Member
Keywords	Cayley-Dickson Construction Octonion Exceptional Lie Algebra g2 Fano Plane Derivation Normed Division Algebra
Date of Defense	2004-04-23
Availability	unrestricted
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.
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