Finite Lorentz transformations, automorphisms, and division algebras.(English)Zbl 0797.53075

Summary: An explicit algebraic description of finite Lorentz transformations of vectors in ten-dimensional Minkowski space is given by means of a parametrization in terms of the octonions. The possible utility of these results for superstring theory is mentioned. Along the way automorphisms of the two highest dimensional normed division algebras, namely, the quaternions and the octonions, are described in terms of conjugation maps. Similar techniques are used to define $$SO(3)$$ and $$SO(7)$$ via conjugation, $$SO(4)$$ via symmetric multiplication, and $$SO(8)$$ via both symmetric multiplication and one-sided multiplication. The noncommutativity and nonassociativity of these division algebras plays a crucial role in our constructions.

MSC:

 53Z05 Applications of differential geometry to physics 17A35 Nonassociative division algebras
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