View source for Quasi-symplectic space

A projective space of odd dimension, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q0767201.png" />, in which the following null-systems (cf. [[Zero system|Zero system]]) are defined:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q0767202.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q0767203.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q0767204.png" /></td> </tr></table>

The first null-system takes points in the space to hyperplanes passing through the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q0767205.png" />-plane

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q0767206.png" /></td> </tr></table>

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

The plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q0767207.png" /> is called the [[Absolute|absolute]], and the two null-systems are absolute null-systems of the quasi-symplectic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q0767208.png" />. A quasi-symplectic space is a special case of a [[Semi-symplectic space|semi-symplectic space]].

Collineations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q0767209.png" /> taking the absolute plane to itself have the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q07672010.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q07672011.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q07672012.png" /></td> </tr></table>

and the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q07672013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q07672014.png" /> are symplectic matrices of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q07672015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q07672016.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q07672017.png" /> is a rectangular matrix with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q07672018.png" /> columns and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q07672019.png" /> rows.

These collineations are called quasi-symplectic transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q07672020.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q07672021.png" /> can be obtained from the symplectic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q07672022.png" /> by limit transition from the absolute of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q07672023.png" /> to the absolute of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076720/q07672024.png" />. 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>