18,990 research outputs found
On the theory of division algebras
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
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
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 and
via conjugation, via symmetric multiplication, and 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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