Alternating 3-Forms and Exceptional Simple Lie Groups of Type G 2

• Published in: Canadian journal of mathematics 1983-10, Vol.35 (5), p.776-806
• Main Author: Herz, Carl
• Format: Article
• Language: English
• ISSN: 0008-414X; 1496-4279
• DOI: 10.4153/CJM-1983-045-2

It is now customary to give concrete descriptions of the exceptional simple Lie groups of type G 2 as groups of automorphisms of the Cayley algebras. There is, however, a more elementary description. Let W be a complex 7-dimensional vector space. Among the alternating 3-forms on W there is a connected dense open subset Ψ( W ) of “maximal” forms. If ψ ∈ Ψ( W ) then the subgroup of AUT C ( W ) consisting of the invertible complex-linear transformations S such that ψ ( S•, S•, S• ) = ψ (•, •, •) is denoted G ( ψ ), and, in Proposition 3.6. we prove where G 1 ( ψ ) is identified with the exceptional simple complex Lie group of dimension 14. Thus the complex Lie algebra of type G 2 is defined in terms of the alternating 3-form ψ alone without the need to specify an invariant quadratic form. In the real case the result is more striking.