The octonions. (English) Zbl 1026.17001

Baez, John C.

Bull. Am. Math. Soc., New Ser. 39, No. 2, 145-205 (2002); errata ibid. 42, 213 (2005).

John Baez exhibits ‘the octonions’! Like in a gallery the reader is encouraged to walk, under competent guidance, from picture to picture, from sculpture to sculpture. Starting with a vivid humorous historic excursion, to Hamilton’s approach to complex numbers and his invention of quaternions, the quite unspectacular birth of octonion theory initiated by Graves and Cayley makes up the first exposition hall.
It is Baez’ ability to be narrative, nearly precise in mathematical terms, but sheltering the reader from technicalities, which makes the reading fun, not only for the mathematically trained but also for a reader with a moderate background.
Having constructed the octonions, much emphasis is laid on a mesh of relations from group theory, homology theory, algebra and representation theory to geometry and especially projective planes. It is this networking which provokes many aha’s and oho’s and unveils a beauty seeable for any open-minded reader. Even physics is touched in subjects like Bott periodicity, triality, Lorentzian geometry, special relativity, string theory and supersymmetry, showing by a heap of examples that ‘the octonions’ is not an idle subject.
This paper encourages anyone to explore further the rich literature given, and even the octonion specialist will find hints in the text where to go for further amusement.

Reviewer: B.Fauser (Tübingen)

MSC:

17-02	Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
17A35	Nonassociative division algebras
17C40	Exceptional Jordan structures
17C90	Applications of Jordan algebras to physics, etc.
22E70	Applications of Lie groups to the sciences; explicit representations

Keywords:

octonions; projective planes; Jordan algebras; exceptional Lie algebras; Clifford algebras; Bott periodicity; Lorentzian geometry; special relativity; supersymmetry; string theory; magic square; Freudenthal triple systems; triality

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