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Oktaven, Ausnahmegruppen und Oktavengeometrie. (German) Zbl 0573.51004
In this article, which is the (first) publication of Freudenthal’s basic lecture notes (written and mimeographed in 1951), the fundamental representations of the real Lie groups \(G_ 2\), \(F_ 4\), and \(E_ 6\) of degrees 7, 26, and 27, respectively, are described from scratch. This is established by introducing first the 8-dimensional real Cayley division algebra and H and \(G_ 2\) as its automorphism group, next the 27-dimensional Jordanalgebra J of Hermitian \(3\times 3\) matrices over H and \(F_ 4\) as its automorphism group, and finally \(E_ 6\) as the group of linear transformations of J preserving a cubic form associated with the algebra structure on J.
The problems stated at the end of the article have mostly been solved, see the introductory note by F. D. Veldkamp in ibid. 19, 3-5 (1985).
Reviewer: A.Cohen

51B25 Lie geometries in nonlinear incidence geometry
20G05 Representation theory for linear algebraic groups
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
17C20 Simple, semisimple Jordan algebras
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