View source for Quasi-symplectic space

<!--
q0767201.png
$#A+1 = 24 n = 0
$#C+1 = 24 : ~/encyclopedia/old_files/data/Q076/Q.0706720 Quasi\AAhsymplectic space
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A projective space of odd dimension,  $  P _ {2n-} 1 $, 
in which the following null-systems (cf. [[Zero system|Zero system]]) are defined:

$$ 
u _ {a}  =  - x  ^ {m+} a ; \ \ 
u _ {m+} a  =  x  ^ {a} ; \ \ 
u _ {m+} b  =  u _ {n+} b  =  0
$$

and

$$ 
u _ {n+} b  =  x  ^ {m+} b ; \ \ 
u _ {m+} b  =  - x  ^ {n+} b ,
$$

$$ 
m  \leq   b  \leq   n - 1 ; \  0  \leq   a  \leq   m - 1 .
$$

The first null-system takes points in the space to hyperplanes passing through the  $  ( 2 n - 2 m - 1 ) $-
plane

$$ 
x  ^ {a}  =  x  ^ {m+} a  =  0 ,
$$

while the second null-system takes points to points of this same plane.

The plane  $  x  ^ {a} = x  ^ {m+} a = 0 $
is called the [[Absolute|absolute]], and the two null-systems are absolute null-systems of the quasi-symplectic space  $  S _ {P _ {2n-} 1 }   ^ {2m-} 1 $. 
A quasi-symplectic space is a special case of a [[Semi-symplectic space|semi-symplectic space]].

Collineations of  $  S _ {P _ {2n-} 1 }   ^ {2m-} 1 $
taking the absolute plane to itself have the form

$$ 
{}  ^  \prime  x  ^ {k}  =  \sum _  \lambda  U _  \lambda   ^ {k} x  ^  \lambda  ,
$$

$$ 
{}  ^  \prime  x  ^ {u}  =  \sum _  \lambda  T _  \lambda   ^ {u} x
 ^  \lambda  + \sum _  \mu  V _  \mu   ^ {u} x  ^  \mu  ,
$$

$$ 
0  \leq   k , \lambda  \leq   2 m - 2 ,\  2 m - 1  \leq   \mu , u  \leq   2 n - 1 ,
$$

and the matrices  $  U _  \lambda   ^ {k} $
and  $  V _  \mu   ^ {u} $
are symplectic matrices of orders  $  2 m $
and  $  2 n - 2 m $; 
$  T _  \lambda   ^ {u} $
is a rectangular matrix with  $  2 m $
columns and  $  2 n - 2 m $
rows.

These collineations are called quasi-symplectic transformations of  $  S _ {P _ {2n-} 1 }   ^ {2m-} 1 $. 
They commute with the given null-systems of the space. The quasi-symplectic invariant of two lines is defined by analogy with the symplectic invariant of lines of a symplectic space.

The quasi-symplectic space  $  S _ {P _ {2n-} 1 }   ^ {2m-} 1 $
can be obtained from the symplectic space  $  S _ {P _ {2n-} 1 }  $
by limit transition from the absolute of  $  S _ {P _ {2n-} 1 }  $
to the absolute of  $  S _ {P _ {2n-} 1 }   ^ {2m-} 1 $. 
Namely, the first of the null-systems given takes all points of the space into planes passing through the absolute plane, while the second takes all planes into points of this same plane.

The quasi-symplectic transformations form a group, which is a Lie group.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.A. Rozenfel'd,   "Non-Euclidean spaces" , Moscow  (1969)  (In Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.A. [B.A. Rozenfel'd] Rosenfel'd,   "A history of non-euclidean geometry" , Springer  (1988)  (Translated from Russian)</TD></TR></table>