Geometric structures on lie groups

The study of geometric structures on manifolds which are locally modelled on homogeneous spaces goes back to Felix Klein`s Erlanger program. Many familiar geometric structures are of this type: classical space forms, flat affine and projective structures, flat conformal structures, spherical CR-structures and many others. An important question which has not been solved up to now is to find a criterion for the existence of such structures on a given manifold or Lie group. In general, these are deep questions. By the work of Klein, and later by Cartan, one can study these geometric structures through algebra. A good example is the case of affine structures on manifolds and left-invariant affine structures on Lie groups. Many results which hold for Euclidean structures can be generalized to this case, at least conjecturally. This has been studied by Milnor and Auslander in the seventies, in connection with affine crystallographic groups and fundamental groups of compact, complete affine manifolds. Since then there has been a lot of progress, also by considering the problem via faithful representations of solvable Lie algebras. However, many questions are still open. In this project we want to continue the study of such geometric structures on Lie groups, and establish the related study of crystallographic actions, simply transitive affine actions of Lie groups, and important generalizations of these. Our methods will be mainly of algebraic nature, using cohomology and representation theory, deformation and degeneration theory, and the study of certain Lie-admissible algebra structures. Some of these algebraic structures have also applications in quantum machanics.

Geometric structures on Lie groups are a central theme in mathematics, because they have connections with many different active research areas. The main goal of this project was to study geometric structures on Lie groups, in particular left- invariant affine structures and their generalizations. The study of such geometric structures has a long history. Felix Klein already studied Euclidean, hyperbolic and spherical geometry in his Erlanger program of 1872 by means of transformation groups which leave the properties of the underlying space invariant. We found new criteria for the existence of geometric structures on given Lie groups. These criteria are mainly of an algebraic nature (in the sense of E. Cartan and W. Thurston). We were able to study affine actions on Lie groups and the resulting generalized geometric structures on the Lie algebra level, by post-Lie algebra structures. We studied not only existence questions, but also the question on the completeness of such structures. We obtained new bounds for the minimal dimension of a faithful module for a given Lie algebra. This is a finite number by Ado`s theorem, the so-called -invariant. It is important, in particular for the existence of geometric structures on Lie groups, but very difficult to determine. Furthermore we obtained related results on periodic derivations and prederivations, Leibniz-derivations and graduations of Lie algebras. We classified Novikov algebras in low dimensions and used this for the study of new covariants, and a complete classification of all orbit closures in the variety of 3-dimensional complex Novikov algebras. Thomas Benes finished his dissertation successfully in this area, with a thesis on "Degenerations of Lie algebras and pre-Lie algebras". Finally we proved new results on the existence of left-invariant affine structures on reductive Lie groups. Felix Behringer is in the course of finishing a Ph. D. thesis on this topic.