Symplectic and Quasisymplectic Geometries

  • Boris Rosenfeld
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 393)

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 −1are called symplectic transformations in this space. The absolute linear complex of Sy 2n −1 can be defined by (2.109), where the (2n × 2n)-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

Authors and Affiliations

  • Boris Rosenfeld
    • 1
  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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