- Doktoratskolleg: Initiativkolleg 1008-N:
**Differential Geometry and Lie Groups.**

Speaker: Peter Michor.

Duration: 01.10.2006-30.09.2009.

Two predoc positions: Wolfgang Moens, Thomas Benes. -
FWF Einzelprojekt P 21683-N13:
**Geometric Structures on Lie Groups.**

Sum: 300K

Duration: 01.10.2009-30.11.2013.

Postdoc position: Wolfgang Moens.

Predoc position: Thomas Benes, Felix Behringer.

In this research project we study geometric structures on manifolds which are locally modelled on homogeneous spaces. Many familiar geometric structures are of this type. An important (unsolved) question is to find a criterion for the existence of such structures on a given manifold or Lie group.

Related topics are crystallographic actions, simply transitive affine actions of Lie groups and important generalizations of these. The methods here are mainly of algebraic nature, using cohomology and representation theory, deformation and degeneration theory, and the study of certain Lie-admissible algebra structures, such as pre-Lie algebra structures.

These algebraic structures have also applications in quantum mechanics. -
FWF Einzelprojekt P 28079-N35:
**Nil-affine Crystallographic Groups and Algebraic Structures.**

Sum: 323K

Duration: 01.01.2016-30.06.2019

Postdoc position: Andrei Minchenko, Wolfgang Moens, Vsevolod Gubarev

Predoc position: Christof Ender

Crystallographic groups have their origin in the study of symmetry groups of crystals in three-dimensional Euclidean space. The theorems of Bieberbach enable us to understand Euclidean crystallographic structures. Since then several other types of crystallographic structures have been considered, e.g., almost-crystallographic and affine crystallographic structures.

The aim of this project is to study so called nil-affine crystallographic structures. These are a natural generalization of affine crystallographic structures, being motivated by open problems on affine structures on manifolds. 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. 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. -
FWO-FWF Internationales Projekt Belgien-Österreich I 3248-N35:
**Automorphisms of Nilpotent Groups and Geometric Structures.**

FWO Project number: G0F9317N, Flemish project leader: Karel Dekimpe

Sum: FWF: 212K and FWO: 267K

Duration: 01.10.2017-2020/21

Postdoc positions: N.N.

The aim of this project is to study automorphism groups of (virtually) nilpotent groups arising in geometry and topology, e.g., in the study of nilmanifolds and infra--nilmanifolds, spectral geometry of Riemannian manifolds, and Nielsen fixed-point theory. So far, automorphism groups in this context have been only studied with ad hoc methods, depending on the specific geometrical or topological problem, which was translated accordingly into certain problem on the automorphism group.

The innovative part of our project lies in the belief that there is a lack of a more general approach, which could be used in several specific situations. Therefore we propose here to start with an in-depth investigation of the automorphism group itself and study several structural aspects of the automorphism group and develop enough computational tools which can then succesfully be applied in a broad range of topological and geometric contexts both by ourselves as by other researchers.

The methods we will use when dealing with nilpotent groups include induction on the nilpotency class, different central series, polynomial functors, and computational algorithms concerning presentations and representations of these groups. Another method is based on a specific eigenvalue study of automorphisms, which can be applied to fixed point theory, expanding maps, Anosov diffeomorphisms, and to the study of the algebraic structure of the the virtually nilpotent group itself, e.g., the property of being co-Hopfian.

Degenerations of pre-Lie algebras, D. Burde, T. Benes,

Journal of Mathematical Physics 50, 112102 (2009).

arXiv:0809.2188v1 (2008).

Complete LR-structures on solvable Lie algebras, D. Burde, K. Dekimpe, K. Vercammen,

Journal of Group Theory 13, Issue 5, 703-719 (2010).

arXiv:0906.1151v1 (2009).

Degenerations of Lie algebras and pre-Lie algebras, T. Benes,

Dissertation, Universität Wien, 28.06.2011.

Abelian ideals of maximal dimension for solvable Lie algebras, D. Burde, C. Ceballos,

Journal of Lie Theory 22, No. 3, 741-756 (2012).

arXiv:0911.2995v2 (2011).

A Characterization of nilpotent Lie algebras by invertible Leibniz-derivations, W. A. Moens,

Communications in Algebra 41, Issue 7, 2427-2440 (2013).

arXiv:1011.6186v1 (2010).

Faithful Lie algebra modules and quotients of the universal enveloping algebra, D. Burde, W. A. Moens,

Journal of Algebra, Vol. 325, Issue 1, 440-460 (2011).

arXiv:1006.2062v1 (2010).

Affine actions on Lie groups and post-Lie algebra structures, D. Burde, K. Dekimpe, K. Vercammen,

Linear Algebra and its Applications 437, 1250-1263 (2012).

arXiv:1109.0251v1 (2011).

Classification of Novikov algebras, D. Burde, W. A. de Graaf,

Applicable Algebra in Engineering, Communication and Computing, Vol. 24, Issue 1,1-15 (2013).

arXiv:1106.5954v1 (2011).

Post-Lie algebra structures and generalized derivations of semisimple Lie algebras, D. Burde, K. Dekimpe,

Moscow Mathematical Journal Vol. 13, Issue 1, 1-18 (2013).

arXiv:1108.5950v1 (2011).

Periodic derivations and prederivations of Lie algebras, D. Burde, W. A. Moens,

Journal of Algebra, Vol. 357, Issue 1, 208-221 (2012).

arXiv:1108.3548v1 (2011).

Classification of orbit closures in the variety of 3-dimensional Novikov algebras, D. Burde, T. Benes,

Journal of Algebra and Its Applications, Vol. 13, Issue 02, 1350081 (2014).

arXiv:1205.5714v1 (2012).

Derived length and nildecomposable Lie algebras, D. Burde,

Scientific Buletin of the "Politehnica" University of Timisoara, 15-24 (2013). ISSN 1224-6069.

arXiv:1212.3113 (2012).

Derivation double Lie algebras, D. Burde,

Journal of Algebra and Its Applications, Vol. 15, Issue 06, 17 pp. (2016).

arXiv:1411.0950v2 (2015).

Post-Lie algebra structures on pairs of Lie algebras, D. Burde, K. Dekimpe,

Journal of Algebra, Vol. 464, 226-245 (2016).

arXiv:1505.00955v2 (2016).

Commutative post-Lie algebra structures on Lie algebras, D. Burde, W. A. Moens,

Journal of Algebra, Vol. 467, 183-201 (2016).

arXiv:1512.05096 (2016).

Calculating Galois groups of third order linear differential equations with parameters, A. Minchenko, A. Ovchinnikov,

Preprint (2016).

arXiv:1611.01784 (2016).

Étale representations for reductive algebraic groups with one-dimensional center, D. Burde, W. Globke,

Journal of Algebra, Vol. 487, 200-216 (2017).

arXiv:1606.01643v2 (2016).

Almost inner derivations of Lie algebras, D. Burde, K. Dekimpe, and B. Verbeke,

To appear in Journal of Algebra and Its Applications (2017).

arXiv:1704.06159 (2017).

Étale representations for reductive algebraic groups with simple factors Sp_n or SO_n, D. Burde, W. Globke,

Preprint (2017).

arXiv:1706.08735 (2017).

Commutative post-Lie algebra structures and linear equations for nilpotent Lie algebras, with K. Dekimpe, and W. A. Moens,

Preprint (2017).

arXiv: 1711.01964 (2017).

Étale representations for reductive algebraic groups with one-dimensional center, D. Burde, W. Globke,

Journal of Algebra, Vol. 487, 200-216 (2017).

arXiv:1606.01643v2 (2016).

Almost inner derivations of Lie algebras, D. Burde, K. Dekimpe, and B. Verbeke,

To appear in Journal of Algebra and Its Applications (2017).

arXiv:1704.06159 (2017).

Étale representations for reductive algebraic groups with simple factors Sp_n or SO_n, D. Burde, W. Globke,

Preprint (2017).

arXiv:1706.08735 (2017).

Commutative post-Lie algebra structures and linear equations for nilpotent Lie algebras, with K. Dekimpe, and W. A. Moens,

Preprint (2017).

arXiv: 1711.01964 (2017)