# Geometry and Algebra

Projective and Cayley-Klein Geometries

Under the influence of my teachers Hans Reichardt and W. Blaschke my mathematical studies have been guided by the ideas of Sophus Lie and Felix Klein, in particular by the Erlanger Program. In the studying year 1961/62 I visited the Moscow State University and listened lectures of P. K. Raschewski, A. L. Onishchik, E. B. Dynkin and others. In this time arose my friendship. with Arkady Onishchik leading to a fruitful cooperation over many years, and finally to our monography

Some remarks about the intentions and the origin of this book one finds here.

For hints, corrections and comments the authors will be thankful. Please send to

Tensor Algebra

In Wolfram's Mathematica v. 11.1.1 exist useful and interesting tensor functions and operations, but there isn't defined a tensor object. The tensors in the examples of the tutorial and the help pages are always Arrays. Concepts like covector, covariant, contravariant tensors, or tensors of mixed type are missing. In my Mathematica notebook Vector and Tensor Algebra (Version 3) these objects are introduced in Mathematica. Given a finite dimensional vector space V over a field K it is shown that the tensors of a fixed type again form a vector space over K. With the tensor product as operation the direct sum of all tensors spaces over V is an associative, non commutative algebra over K. All the algebraic tensor operations: sum, product with a scalar, tensor product, contraction, wedge product for the new tensor objects are implemented into Mathematica. The programming is done in the Mathematica package it must be imported into the notebook before starting interactive working. This package will be the fundament for developing Mathematica applications of tensors in differential geometry and physics. The notebook may serve as a tool accompanying a course or private study to help understanding and applying tensor algebra.

Elementary Affine Geometry

So far I wrote to this subject only a small notebook titled Affine Geometry containing a Mathematica implementation of H. Weyl's point - vector axiomatic s, see his book "Space, Time, Matter", § 1.2. This notebook is not meant to be a systematic introduction to affine geometry. An interested user could take it as a starting point to create a systematic interactive textbook. The notebook may be downloaded here. To work with the notebook one needs the package tensalgv3.m down loadable here. Finished Jan 3,2017.

Elementary Euclidean Geometry

Wolfram's Mathematica contains all linear algebraic tools necessary for working in Euclidean Geometry. The notebook “Euclidean Geometry” shows these tools and some variations of them and applies them to define some functions useful in particular in conformal Euclidean geometry. Here you can see a survey of the contents: Euklidsh.pdf . Obviously, the notebook may not and will not replace a textbook about Euclidean geometry. I hope that it will be useful for students of the first course who want learning linear algebra and geometry and be introduced into Mathematica. You may download the notebook Euklid.nb together with the package vectorcalc.m, containing affine geometric tools, the package euvec.m, containing modules belonging to Euclidean geometry, and a protocol file rank1.txt, here. Finished January 30, 2010

The Subsection Elementary Moebius Geometry is now a part of the Chapter Möbius Geometry.

## Symplectic Geometry

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 the notebook symplectic.nb. 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.

## Lie Algebras

For the classical series of Lie algebras over the real or the complex numbers sometimes extensive calculations of basic invariants like Killing form or index are necessary. Some tools and results in this field are contained in the notebook liealgeb.nb, see further information