Symplectic and Quasisymplectic Geometries

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


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
$$\sum\limits_{i = 0}^{n - 1} {{p^{2i,2i + 1}}} = 0.$$


Linear Complex Jordan Algebra Cross Ratio Symplectic Space Symplectic Transformation 
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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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