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## Preview

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## Copyright information

© Springer Science+Business Media Dordrecht 1997