Quasi-symplectic space

A projective space of odd dimension, $ P _ {2n-} 1 $, in which the following null-systems (cf. 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, 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.

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

[1]	B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)

References

[a1]	B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)