1993/02/12 by Corinne A. Manogue, Jörg Schray · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Algebra representation #Algebraic and Geometric Analysis #Arithmetic #Associative algebra #Associative property #Automorphism #Combinatorics #Commutative property #Division (mathematics) #Division algebra #Lorentz transformation #Mathematical physics #Mathematics #Matrix Theory and Algorithms #Minkowski space #Multiplication (music) #Non-associative algebra #Noncommutative geometry #Physics #Pure mathematics #Quaternion #Vector space #gr-qc #hep-th
paper · pdf · doi:10.1063/1.530056
published as J.Math.Phys. 34 (1993) 3746-3767 · 24 pages, Plain TeX, 2 figures on 1 page submitted separately as uuencoded compressed tar file
arxiv created 1993/02/12 · openalex publication_date 1993/08/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.