24 research outputs found
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
Vector product algebras
Vector products can be defined on spaces of dimensions 0, 1, 3 and 7 only,
and their isomorphism types are determined entirely by their adherent symmetric
bilinear forms. We present a short and elementary proof for this classical
result.Comment: 7 page
The double sign of a real division algebra of finite dimension greater than one
For any real division algebra A of finite dimension greater than one, the
signs of the determinants of left multiplication and right multiplication by a
non-zero element are shown to form an invariant of A, called its double sign.
The double sign causes the category of all real division algebras of a fixed
dimension n>1 to decompose into four blocks. The structures of these blocks are
closely related, and their relationship is made precise for a sample of full
subcategories of the category of all finite-dimensional real division algebras.Comment: 12 page