Constructions for octonion and exceptional Jordan algebras


L. J. Rylands and D. E. Taylor


Research Report 2000-2
Date: 10 January 2000


In this note we reverse the usual process of constructing the Lie algebras of types G2 and F4 as algebras of derivations of the split octonions and the exceptional Jordan algebra and instead begin with their Dynkin diagrams and then construct the algebras together with an action of the Lie algebras and associated Chevalley groups. This is shown to be a variation on a general construction of all standard modules for simple Lie algebras and it is well suited for use in computational algebra systems. All the structure constants which occur are integral and hence the construction specialises to all fields, without restriction on the characteristic, avoiding the usual problems with characteristics 2 and 3.

Key phrases

Exceptional Lie algebras. Jordan algebras. Octonions. Derivation algebras

AMS Subject Classification (1991)

Primary: 17B25
Secondary: 17C40,17D05


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