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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p0734501.png" /> be a set of points with a non-empty collection of distinguished subsets of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p0734502.png" />, called lines. Such a structure is called a polar space if for each line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p0734503.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p0734504.png" /> and each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p0734505.png" /> the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p0734506.png" /> is collinear either with precisely one or with all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p0734507.png" />. 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.
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Examples arise by taking a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p0734508.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p0734509.png" /> to get something non-trivial) with a [[Polarity|polarity]] defined by a non-degenerate bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345010.png" />. Take the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345011.png" /> of absolute points (also called isotropic points), i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345012.png" />. The lines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345013.png" /> are the projective lines of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345014.png" /> which are entirely in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345015.png" />. The name  "polar space"  derives from this class of examples.
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A subspace of a polar space is a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345017.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345020.png" /> are collinear and unequal, then the whole line through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345022.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345023.png" />. A singular subspace of a polar space is one in which every pair of points of it is collinear.
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Let  $  P $
 +
be a set of points with a non-empty collection of distinguished subsets of cardinality  $  \geq  2 $,
 +
called lines. Such a structure is called a polar space if for each line  $  l $
 +
of  $  P $
 +
and each point  $  A \in P \setminus  l $
 +
the point  $  A $
 +
is collinear either with precisely one or with all points of $  l $.  
 +
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.
  
A Tits polar space of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345026.png" />, is a set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345027.png" /> together with a family of subsets, called subspaces, such that:
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Examples arise by taking a projective space $  \mathbf P  ^ {d} $(
 +
$  d \geq  3 $
 +
to get something non-trivial) with a [[Polarity|polarity]] defined by a non-degenerate bilinear form  $  Q $.
 +
Take the subset  $  P $
 +
of absolute points (also called isotropic points), i.e. $  P = \{ {x \in \mathbf P  ^ {d} } : {Q ( x , x ) = 0 } \} $.  
 +
The lines in  $  P $
 +
are the projective lines of  $  \mathbf P  ^ {d} $
 +
which are entirely in  $  P $.  
 +
The name  "polar space" derives from this class of examples.
  
i) a subspace together with the subspaces contained in it is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345028.png" />-dimensional projective space;
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A subspace of a polar space is a subset  $  P  ^  \prime  $
 +
of  $  P $
 +
such that if  $  A, B \in P  ^  \prime  $
 +
and  $  A $
 +
and  $  B $
 +
are collinear and unequal, then the whole line through  $  A $
 +
and  $  B $
 +
is in  $  P  ^  \prime  $.
 +
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  $  n $,
 +
$  n \geq  2 $,
 +
is a set of points  $  P $
 +
together with a family of subsets, called subspaces, such that:
 +
 
 +
i) a subspace together with the subspaces contained in it is a $  d $-
 +
dimensional projective space;
  
 
ii) the intersection of two subspaces is a subspace;
 
ii) the intersection of two subspaces is a subspace;
  
iii) given a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345029.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345030.png" /> and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345031.png" />, there is a unique subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345032.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345033.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345034.png" /> has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345035.png" />; the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345036.png" /> contains all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345037.png" /> that are joined to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345038.png" /> by a line (a subspace of dimension 1);
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iii) given a subspace $  V $
 +
of dimension $  n- 1 $
 +
and a point $  A \in P \setminus  V $,  
 +
there is a unique subspace $  W $
 +
containing $  A $
 +
such that $  V \cap W $
 +
has dimension $  n - 2 $;  
 +
the space $  W $
 +
contains all points of $  V $
 +
that are joined to $  A $
 +
by a line (a subspace of dimension 1);
  
iv) there exist at least two disjoint subspaces of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345039.png" />.
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iv) there exist at least two disjoint subspaces of dimension $  n- 1 $.
  
The Tits polar spaces of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345040.png" /> are known , [[#References|[a2]]] and are classical, i.e. they are Tits polar spaces arising from a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345041.png" />-Hermitian form (cf. [[Sesquilinear form|Sesquilinear form]]) or a [[Pseudo-quadratic form|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345042.png" />). In particular, the subspaces of a finite polar space of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345043.png" /> are the totally-isotropic subspaces with respect to a [[Polarity|polarity]] of a finite projective space or the projective spaces in a non-singular quadric in a finite projective space.
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The Tits polar spaces of rank $  \geq  3 $
 +
are known , [[#References|[a2]]] and are classical, i.e. they are Tits polar spaces arising from a $  ( \sigma - \epsilon ) $-
 +
Hermitian form (cf. [[Sesquilinear form|Sesquilinear form]]) or a [[Pseudo-quadratic form|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 $  \geq  2 $).  
 +
In particular, the subspaces of a finite polar space of rank $  \geq  3 $
 +
are the totally-isotropic subspaces with respect to a [[Polarity|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073450/p07345044.png" />, then the singular subspaces define a classical polar space [[#References|[a3]]].
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Every non-degenerate polar space is linear, and if for a non-degenerate polar space of finite rank all lines have cardinality $  \geq  3 $,  
 +
then the singular subspaces define a classical polar space [[#References|[a3]]].
  
 
A non-degenerate polar space is either classical or a generalized quadrangle (cf. [[Quadrangle, complete|Quadrangle, complete]]).
 
A non-degenerate polar space is either classical or a generalized quadrangle (cf. [[Quadrangle, complete|Quadrangle, complete]]).

Latest revision as of 08:06, 6 June 2020


Let $ P $ be a set of points with a non-empty collection of distinguished subsets of cardinality $ \geq 2 $, called lines. Such a structure is called a polar space if for each line $ l $ of $ P $ and each point $ A \in P \setminus l $ the point $ A $ is collinear either with precisely one or with all points of $ l $. 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 $ \mathbf P ^ {d} $( $ d \geq 3 $ to get something non-trivial) with a polarity defined by a non-degenerate bilinear form $ Q $. Take the subset $ P $ of absolute points (also called isotropic points), i.e. $ P = \{ {x \in \mathbf P ^ {d} } : {Q ( x , x ) = 0 } \} $. The lines in $ P $ are the projective lines of $ \mathbf P ^ {d} $ which are entirely in $ P $. The name "polar space" derives from this class of examples.

A subspace of a polar space is a subset $ P ^ \prime $ of $ P $ such that if $ A, B \in P ^ \prime $ and $ A $ and $ B $ are collinear and unequal, then the whole line through $ A $ and $ B $ is in $ P ^ \prime $. 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 $ n $, $ n \geq 2 $, is a set of points $ P $ together with a family of subsets, called subspaces, such that:

i) a subspace together with the subspaces contained in it is a $ d $- dimensional projective space;

ii) the intersection of two subspaces is a subspace;

iii) given a subspace $ V $ of dimension $ n- 1 $ and a point $ A \in P \setminus V $, there is a unique subspace $ W $ containing $ A $ such that $ V \cap W $ has dimension $ n - 2 $; the space $ W $ contains all points of $ V $ that are joined to $ A $ by a line (a subspace of dimension 1);

iv) there exist at least two disjoint subspaces of dimension $ n- 1 $.

The Tits polar spaces of rank $ \geq 3 $ are known , [a2] and are classical, i.e. they are Tits polar spaces arising from a $ ( \sigma - \epsilon ) $- 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 $ \geq 2 $). In particular, the subspaces of a finite polar space of rank $ \geq 3 $ 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 $ \geq 3 $, 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
How to Cite This Entry:
Polar space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polar_space&oldid=48229