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Éléments de la géométrie des octaves de Cayley. (Thèse). (Foundations of the geometry of Cayley octaves). (French) Zbl 0635.17012

H. Freudenthal investigated the exceptional simple Lie groups and Lie algebras, and the projective plane of Cayley octaves [Oktaven, Ausnahmegruppen und Oktavengeometrie (Math. Inst. Rijksuniv. te Utrecht, 1951; Zbl 0056.259), Neuaufl. (1960; Zbl 0100.030). Also, Geom. Dedicata 19, 7-63 (1985; Zbl 0573.51004), and cf. Indagationes Math. 15, 195-200 (1953; Zbl 0053.015)]. The purpose of this paper is to study the projective plane of octaves due to H. Freudenthal and some other results systematically and algebraically.
The paper, which is the author’s thesis, consists of three parts: the Cayley algebra of octaves, the exceptional Jordan algebra of \(3\times 3\) Hermitian matrices with octaves as coefficients, and the projective plane of Cayley octaves. The topological and differential geometric treatments for the projective plane of octaves are also given.
Reviewer: K.Yamaguti

MSC:

17C40 Exceptional Jordan structures
17B25 Exceptional (super)algebras
51A35 Non-Desarguesian affine and projective planes
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