1,309 research outputs found
A Quasi Curtis-Tits-Phan theorem for the symplectic group
We obtain the symplectic group \SP(V) as the universal completion of an
amalgam of low rank subgroups akin to Levi components. We let \SP(V) act
flag-transitively on the geometry of maximal rank subspaces of . We show
that this geometry and its rank residues are simply connected with few
exceptions. The main exceptional residue is described in some detail. The
amalgamation result is then obtained by applying Tits' lemma. This provides a
new way of recognizing the symplectic groups from a small collection of small
subgroups
A geometric characterization of the classical Lie algebras
A nonzero element x in a Lie algebra g over a field F with Lie product [ , ]
is called a extremal element if [x, [x, g]] is contained in Fx.
Long root elements in classical Lie algebras are examples of extremal
elements. Arjeh Cohen et al. initiated the investigation of Lie algebras
generated by extremal elements in order to provide a geometric characterization
of the classical Lie algebras generated by their long root el- ements. He and
Gabor Ivanyos studied the so-called extremal geometry with as points the
1-dimensional subspaces of g generated by extremal elements of g and as lines
the 2-dimensional subspaces of g all whose nonzero vectors are extremal. For
simple finite dimensional g this geometry turns out to be a root shadow space
of a spherical building. In this paper we show that the isomorphism type of g
is determined by its extremal geometry, provided the building has rank at least
3
Highest weight modules and polarized embeddings of shadow spaces
Let Gamma be the K-shadow space of a spherical building Delta. An embedding V
of Gamma is called polarized if it affords all "singular" hyperplanes of Gamma.
Suppose that Delta is associated to a Chevalley group G. Then Gamma can be
embedded into what we call the Weyl module for G of highest weight lambda_K. It
is proved that this module is polarized and that the associated minimal
polarized embedding is precisely the irreducible G-module of highest weight
lambda_K. In addition a number of general results on polarized embeddings of
shadow spaces are proved. The last few sections are devoted to the study of
specific shadow spaces, notably minuscule weight geometries, polar
grassmannians, and projective flag-grassmannians. The paper is in part
expository in nature so as to make this material accessible to a wide audience.Comment: Improvement in exposition of Sections 1-3 and . Notation improved.
References added. Main results unchange
On the simple connectedness of hyperplane complements in dual polar spaces
Let be a dual polar space of rank \geq 4 be a hyperplane of
and be the complement of \Delta\Delta points, then is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings
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