Polar space
Let be a set of points with a non-empty collection of distinguished subsets of cardinality
, called lines. Such a structure is called a polar space if for each line
of
and each point
the point
is collinear either with precisely one or with all points of
. A non-degenerate polar space is one which has no points that are collinear with other points (i.e. it is not a "cone" ). A polar space is linear if two distinct lines have at most one common point.
Examples arise by taking a projective space (
to get something non-trivial) with a polarity defined by a non-degenerate bilinear form
. Take the subset
of absolute points (also called isotropic points), i.e.
. The lines in
are the projective lines of
which are entirely in
. The name "polar space" derives from this class of examples.
A subspace of a polar space is a subset of
such that if
and
and
are collinear and unequal, then the whole line through
and
is in
. A singular subspace of a polar space is one in which every pair of points of it is collinear.
A Tits polar space of rank ,
, is a set of points
together with a family of subsets, called subspaces, such that:
i) a subspace together with the subspaces contained in it is a -dimensional projective space;
ii) the intersection of two subspaces is a subspace;
iii) given a subspace of dimension
and a point
, there is a unique subspace
containing
such that
has dimension
; the space
contains all points of
that are joined to
by a line (a subspace of dimension 1);
iv) there exist at least two disjoint subspaces of dimension .
The Tits polar spaces of rank are known , [a2] and are classical, i.e. they are Tits polar spaces arising from a
-Hermitian form (cf. Sesquilinear form) or a pseudo-quadratic form on a vector space over a division ring, by taking as subspaces the totally-isotropic subspaces of the form (of Witt index
). In particular, the subspaces of a finite polar space of rank
are the totally-isotropic subspaces with respect to a polarity of a finite projective space or the projective spaces in a non-singular quadric in a finite projective space.
Every non-degenerate polar space is linear, and if for a non-degenerate polar space of finite rank all lines have cardinality , then the singular subspaces define a classical polar space [a3].
A non-degenerate polar space is either classical or a generalized quadrangle (cf. Quadrangle, complete).
References
[a1a] | F.D. Veldkamp, "Polar geometry" Indag. Math. , 21 (1959) pp. 512–551 |
[a1b] | F.D. Veldkamp, "Polar geometry" Indag. Math. , 22 (1960) pp. 207–212 |
[a2] | J. Tits, "Buildings and BN-pairs of spherical type" , Springer (1974) pp. Chapt. 8 |
[a3] | F. Buekenhout, E.E. Shult, "On the foundations of polar geometry" Geom. Dedicata , 3 (1974) pp. 155–170 |
[a4] | R. Dembowski, "Finite geometries" , Springer (1968) pp. 254 |
Polar space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polar_space&oldid=19050