Characterizing symplectic quadrangles by their derivations

Collineations of polar spaces with restricted displacements

“…For instance, for two lines, there are six possible mutual positions given by (1) equality, (2) being contained in a common plane, (3) intersecting in a unique point but not contained in a plane, (4) being disjoint but some plane contains one of them and intersects the other in a point, (5) being disjoint and no plane containing one of them intersects the other in a point, (6) both contained in a common projective subspace, but not in a plane. Clearly, in the cases (2), (3) and (6) the lines are at distance two from each other. But for points, it does.…”

Algebraic Polygons

“…For two nonisomorphic fields K, K Ј, the associated algebraic polygons are not isomorphic. This can be seen as follows: the derived incidence structure of such a polygon is isomorphic to the projective plane w x over the corresponding field, see 18,19 , hence the field can be recovered from the geometry. …”

Half-regular and regular points in compact polygons

“…We will call it the symplectic quadrangle (resp. split Cayley hexagon) over the topological field F. For more details we refer to [12] and [4].…”

“…In Section 4 we treat regular points and study the derived structure in a regular point. Besides an extension of the main theorem of [12] we prove the following (1.1) THEOREM. For a compact connected generalized hexagon S where pointrows and linepencils are manifolds, the following properties are equivalent: (…”

“…The present paper was inspired by [12], where compact symplectic quadrangles were characterized by their derivations, and by [11], where the notion of regular points in generalized hexagons was introduced. We discuss the notion of halfregular and regular points in compact generalized polygons.…”