18,990 research outputs found

    On the theory of division algebras

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    AbstractThe theory of division algebras (of finite dimension over the center) is reduced to an application of two simple principles from general linear algebra. Along the way, a simple proof of Wedderburn's theorem on finite division algebras is given

    Normal forms for the G_2-action on the real symmetric 7x7-matrices by conjugation

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    The exceptional Lie group G_2 acts on the set of real symmetric 7x7-matrices by conjugation. We solve the normal form problem for this group action. In view of earlier results, this gives rise to a classification of all finite-dimensional real flexible division algebras. By a classification is meant a list of pairwise non-isomorphic algebras, exhausting all isomorphism classes. We also give a parametrisation of the set of all real symmetric matrices, based on eigenvalues.Comment: 23 pages. Made typographical update in accordance with the final, published versio

    Finite Lorentz Transformations, Automorphisms, and Division Algebras

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    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)SO(3) and SO(7)SO(7) via conjugation, SO(4)SO(4) via symmetric multiplication, and SO(8)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.Comment: 24 pages, Plain TeX, 2 figures on 1 page submitted separately as uuencoded compressed tar fil
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