Affine geometry on Lie groups and Lie-algebraic structures

This project belongs to the area of algebra and geometry in the field of basic scientific research. In particular, it deals with the connection and interaction of these two areas. When talking about geometry one usually thinks of figures in space, like lines, circles, cubes, spheres, curves, surfaces and so on. When a mathematician is working with these objects then he needs to be able to describe them in a very precise way, so that one can understand their properties and interactions. For this it is necessary that one can compute the objects and their interactions. However, it turns out that it is very difficult to do this only with the geometric objects themselves. In fact, it is impossible to ``compute`` with lines, circles, ellipses etc. directly. To resolve this obstacle, one tries to introduce algebraic systems allowing a translation of the geometric problems into computable algebraic problems. The most well-known system here is the coordination system. Each point in the plane can be described by two numbers, given by the horizontal and vertical position. In the same way, each point in three-space can be described by three numbers and so on. This way one can describe lines, circles, ellipses and other things by algebraic equations and is able to do computations. Of course, one can go much further than just considering a coordinate system. In our project we do not use only numbers, but also more complicated structures, called groups and algebras. They are very well suited to describe problems in geometry and make them computable. We want to study such groups and algebras arising from geometry in detail, in particular Lie groups and Lie algebras, named after the Norwegian mathematician Sophus Lie. The aim is to obtain fundamental results on problems in geometry by studying the associated algebraic structures.

PR-Summary This project investigated problems in the areas of algebra and geometry, with particular emphasis on the connection between these two fields. The Norwegian mathematician Sophus Lie has introduced an algebraic structure, now called "Lie algebras", which can be used to understand and compute geometric structures. In general, it is quite difficult to compute, or to classify geometric structures. The use of algebraic methods was already very sucessful in physics. Symmetry groups, which in physics describe conservation laws and the fundamental laws of nature, are often Lie groups that can be understood through their Lie algebras. In the project, we investigated algebraic structures that naturally generalize such Lie algebra structures and make it possible to obtain results on the existence of certain geometric structures. The question of whether certain geometric structures can exist on given spaces is, in general, very challenging. Furthermore, if such a geometric structure does exist, one would also like to determine all such structures. On these questions, we achieved important results.