Geometry of Lie Groups pp 311-330 | Cite as

# Symplectic and Quasisymplectic Geometries

Chapter

## Abstract

In

**4.1.1**we have seen that the absolutes of the spaces*S*^{ n },*H*^{ n },*S*_{ l }^{ n }, and*H*_{ l }^{ n }are imaginary or real hyperquadrics, which, as we have seen in**2.8.3**are cosymmetry figures in the space*P*^{ n }. In**2.8.3**we have also seen that, besides hyperquadrics, in*P*^{2n+1}there are cosymmetry figures of an other kind: linear complexes of lines. The space*P*^{2n −1}in which a linear complex of lines is given is said to be a*real quadratic symplectic space*and is denoted by*Sy*^{2n −1}. The linear complex determining this space is called the*absolute linear complex*of*Sy*^{2n −1}. The collineations in*P*^{2n −1}preserving the absolute linear complex of*Sy*^{2n −1}are called*symplectic transformations*in this space. The absolute linear complex of*Sy*^{2n −1}can be defined by (2.109), where the (2*n*× 2*n*)-matrix (*a*_{ ij }) can be reduced to the form (0.62); then (2.109) has the form$$\sum\limits_{i = 0}^{n - 1} {{p^{2i,2i + 1}}} = 0.$$

(6.1)

## Keywords

Linear Complex Jordan Algebra Cross Ratio Symplectic Space Symplectic Transformation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media Dordrecht 1997