Constructions for Octonion and Exceptional Jordan Algebras

  • Authors:

  • L. J. Rylands;D. E. Taylor

  • Affiliations:

  • School of Science, University of Western Sydney, Nepean, Australia;School of Mathematics and Statistics, University of Sydney, NSW 2006 Australia

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2000

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Abstract

In this note we reverse theusual process of constructing the Lie algebras of types G_2and F_4 as algebras of derivations of the splitoctonions or the exceptional Jordan algebra and instead beginwith their Dynkin diagrams and then construct the algebras togetherwith an action of the Lie algebras and associated Chevalley groups.This is shown to be a variation on a general construction ofall standard modules for simple Lie algebras and it is well suitedfor use in computational algebra systems. All the structure constantswhich occur are integral and hence the construction specialisesto all fields, without restriction on the characteristic, avoidingthe usual problems with characteristics 2 and 3.