Title page for ETD etd-04232004-131642

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
  • Cayley-Dickson Construction
  • Octonion
  • Exceptional Lie Algebra g2
  • Fano Plane
  • Derivation
  • Normed Division Algebra
Date of Defense 2004-04-23
Availability unrestricted
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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