- 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: 01.10.2017-30.09.2022

Postdoc positions: Wolfgang Globke, Kaidi Ye, Marcos Origlia, Wolfgang Moens

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. -
FWF Einzelprojekt P 33811-N:
**Affine Geometry on Lie Groups and Lie-algebraic Structures.**

Sum: 349K

Duration: 01.01.2021-31.12.2025

Postdoc position: Pilar Páez-Guillán, Simone Blumer

Predoc position: Mina Monadjem

The goal of this project is to study affine and nil-affine geometry on Lie groups and its related algebraic structures by means of Lie-algebraic methods. This includes, among other things, nil-affine actions on Lie groups and crystallographic actions. The associated algebraic structures are Lie algebras, pre-Lie algebras, post-Lie algebras, prehomogeneous modules, classical and modified Yang-Baxter equations and Rota-Baxter operators.

We have a list of important research questions of different levels in the project description, ranging from the very general and difficult existence questions for geometric structures on Lie groups to the more specific questions on pre-Lie and post-Lie algebras, Yang-Baxter equations, Rota-Baxter operators, decompositions of Lie algebras, cohomology of Lie algebras, faithful modules, almost inner derivations and etale affine representations of reductive algebraic groups. For example, we want to study post-Lie algebra structures on pairs of Lie algebras with complete Lie algebras via the bijective correspondence to Rota-Baxter operators. We ask for generalizations of commutative post-Lie algebra structures. We study rigidity questions for pairs of Lie algebras admitting a post-Lie algebra structure, and want to find out the structure of Lie algebras which are the vector space sum of two semisimple subalgebras. Such algebras must be perfect, but need not be semisimple.

### Project Publications P 21683-N13:

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).

### Project Publications P 28079-N35:

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 differential Galois groups of parametrized differential equations, with applications to hypertranscendence,

C. Hardouin, A. Minchenko, and A. Ovchinnikov,

Mathematische Annalen, Vol. 368, Issue 1-2, 587-632 (2017).

arXiv:1505.07068 (2016).

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

Communications in Contemporary Mathematics, Vol. 20, Issue 4, 1750038, 25 pp (2018).

arXiv:1611.01784 (2016).

Triviality of differential Galois cohomologies of linear differential algebraic groups, A. Minchenko and A. Ovchinnikov,

Communications in Algebra, Vol. 47, Issue 12, 5094-5100 (2019).

arXiv:1707.08620 (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,

Journal of Algebra and Its Applications Vol. 17, Issue 11, 26 pp. (2018).

arXiv:1704.06159 (2017).

Étale representations for reductive algebraic groups with simple factors \(Sp(n)\) or \(SO(n)\), D. Burde, W. Globke, and A. Minchenko,

Transformation Groups, Vol. 24, Issue 3, 769-780 (2019).

arXiv:1706.08735 (2017).

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

Journal of Algebra, Vol. 526, 12-29 (2019).

arXiv: 1711.01964 (2017).

Post-Lie algebra structures for nilpotent Lie algebras, D. Burde, C. Ender, and W. A. Moens,

International Journal of Algebra and Computation, Vol. 28, Issue 5, 915-933 (2018).

arXiv: 1801.05652 (2018)

Rota-Baxter Operators on Quadratic Algebras, P. Benito, V. Gubarev, and A. Pozhidaev.

Mediterranean Journal of Mathematics, Vol. 15, Issue 5, Art. 189, 23 pages (2018).

arXiv: 1801.07037 (2018)

Rota-Baxter operators and post-Lie algebra structures on semisimple Lie algebras, D. Burde, V. Gubarev,

Communications in Algebra, Vol. 47, Issue 5, 2280-2296 (2019).

arXiv: 1805.05104 (2018)

Commutative post-Lie algebra structures on Kac-Moody algebras, D. Burde, P. Zusmanovich,

Communications in Algebra, Vol. 47, Issue 12, 5218-5226 (2019).

arXiv: 1805.04267 (2018)

Rota-Baxter operators on unital algebras, V. Gubarev,

Moscow Mathematical Journal Vol. 21, Issue 2, 325-364 (2021).

arXiv: 1805.00723 (2018)

Embedding of post-Lie algebras into postassociative algebras, V. Gubarev,

New Trends in Algebra and Combinatorics. Proceedings of the 3rd International Congress in

Algebras and Combinatorics (ICAC2017), World Scientific, Singapore; ISBN 978-981-121-546-9, p. 57-67 (2020).

arXiv: 1808.08839 (2018)

Embedding of pre-Lie algebras into preassociative algebras, V. Gubarev,

Algebra Colloquium, Vol. 27, No. 2, 299-310. (2020).

arXiv: 1808.09822 (2018)

Rota-Baxter operators and Bernoulli polynomials, V. Gubarev,

Communications in Mathematics Vol. 29, Issue 1, 1-14 (2021).

arXiv: 1810.05455 (2018)

Poincare-Birkhoff-Witt theorem for pre-Lie and post-Lie algebras, V. Gubarev,

Journal of Lie theory, Vol. 30, No. 1, 223-238 (2020).

arXiv: 1903.02960 (2019)

Commutative Post-Lie algebra structures on nilpotent Lie algebras and Poisson algebras, D. Burde, C. Ender,

Linear Algebra and its Applications, Vol. 584, 107-126 (2020).

arXiv: 1903.00291 (2019)

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

International Journal of Algebra and Computation, Vol. 31, Issue 02, 341-364 (2021).

arXiv:1905.08145 (2019).

Decompositions of algebras and post-associative algebra structures, D. Burde, V. Gubarev,

International Journal of Algebra and Computation, Vol. 30, Issue 3, 451-466 (2020).

arXiv: 1906.09854 (2019)

### Project Publications I 3248-N35:

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,

Journal of Algebra and Its Applications Vol. 17, Issue 11, 26 pp. (2018).

arXiv:1704.06159 (2017).

Étale representations for reductive algebraic groups with simple factors \(Sp(n)\) or \(SO(n)\), D. Burde, W. Globke, A. Minchenko

Transformation Groups, Vol. 24, Issue 3, 769-780 (2019).

arXiv:1706.08735 (2017).

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

Journal of Algebra, Vol. 526, 12-29 (2019).

arXiv: 1711.01964 (2017)

Post-Lie algebra structures for nilpotent Lie algebras, D. Burde, C. Ender, and W. A. Moens,

International Journal of Algebra and Computation, Vol. 28, Issue 5, 915-933 (2018).

arXiv: 1801.05652 (2018)

Rota-Baxter operators and post-Lie algebra structures on semisimple Lie algebras, D. Burde, V. Gubarev,

Communications in Algebra, Vol. 47, Issue 5, 2280-2296 (2019).

arXiv: 1805.05104 (2018)

Commutative post-Lie algebra structures on Kac-Moody algebras, D. Burde, P. Zusmanovich,

Communications in Algebra, Vol. 47, Issue 12, 5218-5226 (2019).

arXiv: 1805.04267 (2018)

Simply connected indefinite homogeneous spaces of finite volume, O. Baues, W. Globke, A. Zeghib,

arXiv: 1807.02430 (2018)

Isometry Lie algebras of indefinite homogeneous spaces of finite volume, O. Baues, W. Globke, A. Zeghib,

Proceedings of the London Mathematical Society, Vol. 119, Issue 4, 867-1148 (2019).

arXiv: 1803.10436 (2018)

On compact homogeneous \(G_{2(2)}\)-manifolds, W. Globke,

Journal of the Australian Mathematical Society, Vol. 110, no. 1, 71-80 (2021).

arXiv: 1808.10160 (2018)

Commutative Post-Lie algebra structures on nilpotent Lie algebras and Poisson algebras, D. Burde, C. Ender,

Linear Algebra and its Applications, Vol. 584, 107-126 (2020).

arXiv: 1903.00291 (2019)

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

International Journal of Algebra and Computation, Vol. 31, Issue 02, 341-364 (2021).

arXiv:1905.08145 (2019)

Decompositions of algebras and post-associative algebra structures, D. Burde, V. Gubarev,

International Journal of Algebra and Computation, Vol. 30, Issue 3, 451-466 (2020).

arXiv: 1906.09854 (2019)

Crystallographic actions on Lie groups and post-Lie algebra structures, D. Burde,

Communications of Mathematics, Vol. 29, Issue 01, 67-89 (2021).

arXiv:1909.02797 (2019)

A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras, D. Burde, W. A. Moens,

Journal of Algebra, Vol. 559, 529-546 (2020).

arXiv: 2001.04822 (2020)

Almost inner derivations of 2-step nilpotent Lie algebras of genus 2, D. Burde, K. Dekimpe, B. Verbeke,

Linear Algebra and its Applications, Vol. 608, 185-202 (2021).

arXiv:2004.10567 (2020)

The structure of Lie algebras admitting a derivation satisfying a polynomial identity, D. Burde, W. A. Moens,

Communications in Algebra, Vol. 50, Issue 11, 4636-4647 (2022).

arXiv:2009.05434 (2021).

Rigidity of pseudo-Hermitian homogeneous spaces of finite volume, O. Baues, W. Globke, A. Zeghib,

arXiv: 2006.05780 (2020)

A note on étale representations from nilpotent orbits, H. Dietrich, W. Globke, M. Origlia,

Bull. Aust. Math. Soc. 106 (2022), Issue 1, 113-125.

arXiv: 2102.13163 (2021)

Information geometry and asymptotic geodesics on the space of normal distributions, W. Globke, R. Quiroga-Barranco,

Information Geometry Vol. 4, Issue 01, 131-153 (2021).

arXiv: 1904.12921 (2019)

Semisimple decompositions of Lie algebras and prehomogeneous modules, D. Burde, W. A. Moens,

Journal of Algebra, Vol. 604, 664-681 (2022).

arXiv:2201.08758 (2022).

Finite groups admitting a coprime automorphism satisfying an additional identity, W. A. Moens,

Journal of Group Theory, Vol. 26, Issue 2, 357-375 (2023).

arXiv: 2202.10087 (2022)

Rigidity results for Lie algebras admitting a post-Lie algebra structure D. Burde, K. Dekimpe, M. Monadjem,

International Journal of Algebra and Computation, Vol. 32, Issue 08, 1495-1511 (2022).

arXiv:2205.04218 (2022).

Sympathetic Lie algebras and adjoint cohomology for Lie algebras, D. Burde, F. Wagemann,

Journal of Algebra, Vol. 629, 381-398 (2023).

arXiv:1908.05963v3 (2022).

Counterexamples to the Zassenhaus conjecture on simple modular Lie algebras, D. Burde, W. A. Moens and P. Páez-Guillán,

Journal of Algebra, Vol. 629, 21-37 (2023).

arXiv:2209.14822v2 (2022).

### Project Publications P 33811:

The structure of Lie algebras admitting a derivation satisfying a polynomial identity, D. Burde, W. A. Moens,

Communications in Algebra, Vol. 50, Issue 11, 4636-4647 (2022).

arXiv:2009.05434 (2021).

One-generated nilpotent bicommutative algebras, I. Kaygorodov, P. Páez-Guillán, V. Voronin,

Algebra Colloquium, Vol. 29, Issue 3, 453-474 (2022).

arXiv:2112.00417 (2022).

Semisimple decompositions of Lie algebras and prehomogeneous modules, D. Burde, W. A. Moens,

Journal of Algebra, Vol. 604, 664-681 (2022).

arXiv:2201.08758 (2022).

Rigidity results for Lie algebras admitting a post-Lie algebra structure D. Burde, K. Dekimpe, M. Monadjem,

International Journal of Algebra and Computation, Vol. 32, Issue 08, 1495-1511 (2022).

arXiv:2205.04218 (2022).

Sympathetic Lie algebras and adjoint cohomology for Lie algebras, D. Burde, F. Wagemann,

Journal of Algebra, Vol. 629, 381-398 (2023).

arXiv:1908.05963v3 (2022).

Counterexamples to the Zassenhaus conjecture on simple modular Lie algebras, D. Burde, W. A. Moens and P. Páez-Guillán,

Journal of Algebra, Vol. 629, 21-37 (2023).

arXiv:2209.14822v2 (2022).

On the subalgebra lattice of a restricted Lie algebra, P. Páez-Guillán, S. Siciliano and D. A. Towers,

Linear Algebra and its Applications, Vol. 660, 47-65 (2023).

arXiv:2201.01718 (2022).

Central extensions of axial algebras, I. Kaygorodov, C. M. Gonzáles, P. Páez-Guillán,

arXiv:2201.01718 (2022).

Modularity conditions in Leibniz algebras, P. Páez-Guillán, S. Siciliano, D. A. Towers,

To appear in Journal of Algebra and Its Applications (2024).

arXiv:2305.15530 (2023).

Post-Lie algebra structures for perfect Lie algebras, D. Burde, K. Dekimpe and M. Monadjem,

To appear in Communications in Algebra (2024).

arXiv:2311.08985 (2023).