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Alternating 3-Forms and Exceptional Simple Lie Groups of Type G2

Published online by Cambridge University Press:  20 November 2018

Carl Herz
Carl Herz*
Affiliation:
McGill University, Montreal, Quebec
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It is now customary to give concrete descriptions of the exceptional simple Lie groups of type G2 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 AUTC(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 G1(ψ) is identified with the exceptional simple complex Lie group of dimension 14. Thus the complex Lie algebra of type G2 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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 5 , 01 October 1983 , pp. 776 - 806
Copyright
Copyright © Canadian Mathematical Society 1983

References

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