377 research outputs found
Ree geometries
We introduce a rank 3 geometry for any Ree group over a not necessarily perfect field and show that its full collineation group is the automorphism group of the corresponding Ree group. A similar result holds for two rank 2 geometries obtained as a truncation of this rank 3 geometry. As an application, we show that a polarity in any Moufang generalized hexagon is unambiguously determined by its set of absolute points, or equivalently, its set of absolute lines
Some results on spreads and ovoids
We survey some results on ovoids and spreads of finite polar spaces, focusing on the ovoids of H(3,q^2) arising from spreads of PG(3,q)
via indicator sets and Shult embedding, and on some related constructions.
We conclude with a remark on symplectic spreads of PG(2n-1,q)
Is there a Jordan geometry underlying quantum physics?
There have been several propositions for a geometric and essentially
non-linear formulation of quantum mechanics. From a purely mathematical point
of view, the point of view of Jordan algebra theory might give new strength to
such approaches: there is a ``Jordan geometry'' belonging to the Jordan part of
the algebra of observables, in the same way as Lie groups belong to the Lie
part. Both the Lie geometry and the Jordan geometry are well-adapted to
describe certain features of quantum theory. We concentrate here on the
mathematical description of the Jordan geometry and raise some questions
concerning possible relations with foundational issues of quantum theory.Comment: 30 page
Moufang sets arising from polarities of Moufang planes over octonion division algebras
For every octonion division algebra O, there exists a projective plane which is parametrized by O; these planes are related to rank two forms of linear algebraic groups of absolute type E6. We study all possible polarities of such octonion planes having absolute points, and their corresponding Moufang set.
It turns out that there are four different types of polarities, giving rise to (1) Moufang sets of type F4, (2) Moufang sets of type 2E6, (3) hermitian Moufang sets of type C4, and (4) projective Moufang sets over a 5-dimensional subspace of an octonion division algebra.
Case (3) only occurs over fields of characteristic different from two, whereas case (4) only occurs over fields of characteristic equal to two. The Moufang sets of type 2E6 that we obtain in case (2) are exactly those corresponding to linear algebraic groups of type 2E6,1^29; the explicit description of those Moufang sets was not yet known
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