introduction to the notebook symplectic.nb

It is well known that symplectic scalar products, i. e. non degenerated skew symmetric bilinear forms, exist only in even-dimensional vector spaces. A vector space is called symplectic, if a symplectic scalar product for its vectors is distinguished. The symplectic group is defined as the group of linear transformations of the vector space preserving the symplectic scalar product; it acts transitively on the corresponding odd-dimensional projective space, defining the projective symplectic geometry as the theory of geometric properties invariant under this action, in the sense of F. Klein's Erlanger Programm. Since in dimension 2 the symplectic group coincides with the special linear group, the symplectic geometry of a projective line coincides with its projective geometry. The first interesting case is the three-dimensional one, the main subject of this notebook. Closely related to the projective geometry is the spherical geometry. One easily verifies, that the symplectic groups act transitively also on the odd-dimensional spheres, being double coverings of the projective spaces of the same dimension. The tools developed in this notebook can also be applied to explore the spherical symplectic geometries.

In the first section basic concepts of symplectic linear algebra are presented. An algorithm called symplectic orthogonalization seems to be new: it constructs for a given sequence of vectors a symplectic or optional an orthosymplectic sequence of vectors with the same span. This also gives a method to define adapted bases for subspaces. Rank and index of the scalar product restricted to the subspace are calculated, and the symplectic vector sequence consists of a basis of the defect subspace and a symplectic basis for a complementary symplectic subspace within the span.

The second section contains some considerations of symplectic transformations. In particular, we introduce a very simple class of these transformations, namely the symplectic transvections, which generate the symplectic group.

Section 3 is devoted to symplectic line geometry. The absolute of projective symplectic geometry is the complex of the isotropic lines, called the absolute null system. The complement of the nullsystem is the set of symplectic lines, on which the restriction of the scalar product does not vanish. (Remember that the projective lines are the two-dimensional vector subspaces.) For pairs of symplectic lines there exists a symplectic invariant being similar to the distance of two lines in metric geometries. The value of this invariant is the function sym defined and studied in this section.

The aim of the last two sections is the classification of the quadrics in the 3-dimensional complex or real projective symplectic spaces. As in Euclidean or affine geometries one classifies the symmetric bilinear forms whose corresponding quadratic forms define the quadrics. They are equivariantly associated with a special class of endomorphisms, the skew symmetric operators, of the underlying vector space. These operators can be classified using their Jordan decompositions. In section 4 this is done for complex symplectic spaces; for each class normal forms of the operators and the bilinear forms are found. Section 5 describes the refinements necessary for real spaces. Also in this case normal forms are obtained with the help of which one may discuss the shape of the quadrics.

The appendix contains modules useful for any application of Mathematica to linear algebra and the corresponding geometries. They are contained in the package vectorcalc.m. The modules specific for symplectic linear algebra are collected in the package symplecticgeo.m. They may be downloaded from this homepage.

Finished January 9, 2006.