Finite Lorentz Transformations, Automorphisms, and Division Algebras

Details Finite Lorentz Transformations, Automorphisms, and Division Algebras Finite Lorentz Transformations, Automorphisms, and Division Algebras

1993-02-12

arxiv.org/abs/hep-th/9302044v2

J.Math.Phys. 34 (1993) 3746-3767

Physics

High Energy Physics

High Energy Physics - Theory

24 pages, Plain TeX, 2 figures on 1 page submitted separately as uuencoded compressed tar file

Scientific paper

10.1063/1.530056

We give an explicit algebraic description of finite Lorentz transformations of vectors in 10-dimensional Minkowski space by means of a parameterization in terms of the octonions. The possible utility of these results for superstring theory is mentioned. Along the way we describe automorphisms of the two highest dimensional normed division algebras, namely the quaternions and the octonions, in terms of conjugation maps. We use similar techniques 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 non-commutativity and non-associativity of these division algebras plays a crucial role in our constructions.

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