Nil-affine crystallographic groups and algebraic structures

Crystallographic groups have their origin in the study of symmetry groups of crystals in three-dimensional Euclidean space. They have been investigated already hundred years ago. Since then Euclidean crystallographic structures are well understood, and several other types of crystallographic structures have been considered, such as almost- crystallographic and affine crystallographic structures. For the affine case it was expected that the results from the Euclidean case should generalize in a straightforward manner. This, however, turned out to be not the case. Since more than 30 years there is now an active research on affine crystallographic structures. Although there have been obtained many new results, some of the expected generalizations are still open conjectures. The aim of this project is to study so called nil-affine crystallographic structures. These are a very natural generalization of affine crystallographic structures, being motivated by open problems on affine structures. Main progress in the study of affine crystallographic groups has been obtained by using the close relationship to simply transitive affine actions on Lie groups. These actions are in a one-to-one correspondence to certain Lie algebraic structures, which can be treated successfully by means of algebra. Our first aim is, to establish a similar correspondence in the nil-affine case. The algebraic structures arising here are so called post-Lie algebra structures on pairs of Lie algebras. Then we want to study these algebraic structures, in order to obtain structure results or even a classification. Finally also geometric problems on the associated nil-affine manifolds and their fundamental groups will be considered. Our investigations naturally have certain group-theoretical and number-theoretical aspects. The algebraic structures arising here are also of interest in other areas, such as operad theory and theoretical physics, in connection with renormalizable quantum field theories.

Summary for public relations purposes: Nil-affine crystallographic groups and algebraic structures Crystallographic groups arise in the study of symmetry groups of crystals in three-dimensional Euclidean space. They have been studied intensively since more than hundred years. In dimension two they are called "wallpaper groups". In 1900 the German mathematician David Hilbert published a list of 23 unsolved problems at the international congress of mathematics in Paris. The first part of the eighteenth problem addresses the question whether or not there are only finitely many different crystallographic groups in each dimension. Ludwig Bieberbach gave a positive answer in 1910. His theorems, called "Bieberbach Theorems", are part of the foundation of crystallographic groups up to now. The research on this topic has developed since then rapidly and several natural generalizations of crystallographic groups have been considered. This includes affine-crystallographic and nil-affine crystallographic groups, which is the topic of this research project.The main results of our project are as follows. First of all we provide a method which allows us to reduce the study of nil-affine crystallographic groups to certain Lie-algebraic structures, namely to "post-Lie algebra structures". This enables us to obtain several existence and classification results on nil-affine crystallographic structures. Actually, the algebraic methods are here more effective than geometric or topological methods. The study of discrete groups is done here on the level of Lie algebras and Lie-algebraic structures. The existence of post-Lie algebra structures depends very much on the algebraic properties of the underlying pairs of Lie algebras $(G,N)$. Such properties are, for example, simplicity, semisimplicity, reductiveness, perfectness, solvability, nilpotence and other properties. Suppose that G is simple. Then the existence of such structures is only possible if N is isomorphic to G. On the other hand, if N is simple there may exist such structures without G being isomorphic to N. So the situation is not symmetric for the pair of Lie algebras. In general, the existence question is a hard problem and there are several different cases. We can solve here most cases, but not all of them. The classification of these structures is even more complicated and usually only possible if one of the Lie algebras is semisimple. In this case we obtain several interesting classification results. Even in the solvable and nilpotent case, which is hopeless in general, we manage to obtain some classification results. We find several interesting links to other research topics, e.g., we establish a one-to-one correspondence of post-Lie algebra structures in certain cases to Rota-Baxter operators of weight one on N. In this case we can use the theory of such operators for our purposes. We establish further connections to etale affine representations of reductive algebraic groups.