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Karel Dekimpe

Publications and source records attributed to Karel Dekimpe.

51 records · Page 3Linked to original sources
arXiv2014

Existence of Anosov diffeomorphisms on infra-nilmanifolds modeled on free nilpotent Lie groups

An infra-nilmanifold is a manifold which is constructed as a quotient space $Γ\backslash G$ of a simply connected nilpotent Lie group $G$, where $Γ$ is a discrete group acting properly discontinuously and cocompactly on $G$ via so called affine maps. The manifold $Γ\backslash G$ is said to be modeled on the Lie group $G$. This class of manifolds is conjectured to be the only class of closed manifolds allowing an Anosov diffeomorphism. However, it is far from obvious which of these infra--nilmanifolds actually do admit an Anosov diffeomorphism. In this paper we completely solve this question for infra-nilmanifolds modeled on a free $c$--step nilpotent Lie group.

math.DS↗
arXiv2014

The $R_{\infty} property for abelian groups

It is well known there is no finitely generated abelian group which has the $R_\infty$ property. We will show that also many non-finitely generated abelian groups do not have the $R_\infty$ property, but this does not hold for all of them. In fact we construct an uncountable number of infinite countable abelian groups which do have the $R_{\infty}$ property. We also construct an abelian group such that the cardinality of the Reidemeister classes is uncountable for any automorphism of that group. 8 pages, no figures

math.GR↗
arXiv2013

Nielsen zeta functions for maps on infra-nilmanifolds are rational

In this paper we will show that for any map $f$ on an infra-nilmanifold, the Nielsen number $N(f)$ of this map is either equal to $|L(f)|$, where $L(f)$ is the Lefschetz number of that map, or equal to the expression $|L(f)-L(f_+)|$, where $f_+$ is a lift of $f$ to a 2-fold covering of that infra-nilmanifold. By exploiting the exact nature of this relationship for all powers of $f$, we prove that the Nielsen dynamical zeta function for a map on an infra-nilmanifold is always a rational function.

math.AT↗
arXiv2012

An explicit seven-term exact sequence for the cohomology of a Lie algebra extension

We construct a seven-term exact sequence involving low degree cohomology spaces of a Lie algebra $\Lg$, an ideal $\Lh$ of $\Lg$ and the quotient $\Lg / \Lh$ with coefficients in a $\Lg$-module. The existence of such a sequence follows from the Hochschild-Serre spectral sequence associated to the Lie algebra extension. However, some of the maps occurring in this induced sequence are not always explicitly known or easy to describe. In this article, we give alternative maps that yield an exact sequence of the same form, making use of the interpretations of the low-dimensional cohomology spaces in terms of derivations, extensions etc. The maps are constructed using elementary methods. Although we don't know whether the new maps coincide with the ones induced by the spectral sequence, the alternative sequence can certainly be useful, especially since we include straight-forward cocycle descriptions of the constructed maps.

math.RA↗
arXiv2012

A seven-term exact sequence for the cohomology of a group extension

In this paper, we construct a seven-term exact sequence involving the cohomology groups of a group extension. Although the existence of such a sequence can be derived using spectral sequence arguments, there is little knowledge about some of the maps occuring in the sequence, limiting its usefulness. Here we present a construction using only very elementary tools, always related to the notion of conjugation in a group. This results in a complete and usable description of all the maps, which we describe both on cocycle level as on the level of the interpretations of low dimensional cohomology groups (e.g. group extensions).

math.GR↗
arXiv2011

Affine actions on Lie groups and post-Lie algebra structures

We introduce post-Lie algebra structures on pairs of Lie algebras $(\Lg,\Ln)$ defined on a fixed vector space $V$. Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtain several results on the existence of post-Lie algebra structures, in terms of the algebraic structure of the two Lie algebras $\Lg$ and $\Ln$. One result is, for example, that if there exists a post-Lie algebra structure on $(\Lg,\Ln)$, where $\Lg$ is nilpotent, then $\Ln$ must be solvable. Furthermore special cases and examples are given. This includes a classification of all complex, two-dimensional post-Lie algebras.

math.RA↗
arXiv2011

Post-Lie algebra structures and generalized derivations of semisimple Lie algebras

We study post-Lie algebra structures on pairs of Lie algebras (g,n), and prove existence results for the case that one of the Lie algebras is semisimple. For semisimple g and solvable n we show that there exist no post-Lie algebra structures on (g,n). For semisimple n and certain solvable g we construct canonical post-Lie algebra structures. On the other hand we prove that there are no post-Lie algebra structures for semisimple n and solvable, unimodular g. We also determine the generalized $(\al,\be,\ga)$-derivations of n in the semisimple case. As an application we classify post-Lie algebra structures induced by generalized derivations.

math.RA↗
arXiv2011

What an infra-nilmanifold endomorphism really should be

Infra-nilmanifold endomorphisms were introduced in the late sixties. They play a very crucial role in dynamics, especially when studying expanding maps and Anosov diffeomorphisms. However, in this note we will explain that the two main results in this area are based on a false result and that although we can repair one of these two theorems, there remains doubt on the correctness of the other one. Moreover, we will also show that the notion of an infra-nilmanifold endomorphism itself has not always been interpreted in the same way. Finally, we define a slightly more general concept of the notion of an infra-nilmanifold endomorphism and explain why this is really the right concept to work with.

math.DS↗
arXiv2009

Complete LR-structures on solvable Lie algebras

An LR-structure on a Lie algebra is a bilinear product, satisfying certain commutativity relations, and which is compatible with the Lie product. LR-structures arise in the study of simply transitive affine actions on Lie groups. In particular one is interested in the question which Lie algebras admit a complete LR-structure. In this paper we show that a Lie algebra admits a complete LR-structure if and only if it admits any LR-structure.

math.RA↗
arXiv2008

LR-algebras

In the study of NIL-affine actions on nilpotent Lie groups we introduced so called LR-structures on Lie algebras. The aim of this paper is to consider the existence question of LR-structures, and to start a structure theory of LR-algebras. We show that any Lie algebra admitting an LR-structure is 2-step solvable. Conversely we find several classes of 2-step solvable Lie algebras admitting an LR-structure, but also classes not admitting such a structure. We study also ideals in LR-algebras, and classify low-dimensional real LR-algebras.

math.RA↗
arXiv2007

Affine actions on Nilpotent Lie groups

To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence question of such a simply transitive affine action to a corresponding question on the Lie algebra level. As an example of the possible use of this translation, we then consider the case where dim(G)=dim(N) less than 6. Finally, we specialize to the case of abelian simply transitive affine actions on a given connected and simply connected nilpotent Lie group. It turns out that such a simply transitive abelian affine action on N corresponds to a particular Lie compatible bilinear product on the Lie algebra of N, which we call an LR-structure.

math.DG↗
arXiv2007

Novikov algebras and Novikov structures on Lie algebras

We study ideals of Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We present the first example of a three-step nilpotent Lie algebra which does not admit a Novikov structure. On the other hand we show that any free three-step nilpotent Lie algebra admits a Novikov structure. We study the existence question also for Lie algebras of triangular matrices. Finally we show that there are families of Lie algebras of arbitrary high solvability class which admit Novikov structures.

math.RA↗
arXiv2005

Novikov structures on solvable Lie algebras

We study Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov structure must be solvable. Conversely we present an example of a nilpotent 2-step solvable Lie algebra without any Novikov structure. We construct Novikov structures on certain Lie algebras via classical r-matrices and via extensions. In the latter case we lift Novikov structures on an abelian Lie algebra A and a Lie algebra B to certain extensions of B by A. We apply this to prove the existence of affine and Novikov structures on several classes of 2-step solvable Lie algebras. In particular we generalize a well known result of Scheuneman concerning affine structures on 3-step nilpotent Lie algebras.

math-ph↗
arXiv2004

The Auslander conjecture for NIL-affine crystallographic groups

Let N be a simply connected, connected real nilpotent Lie group of finite dimension n. We study subgroups $Γ$ in $\Aff (N)=N\rtimes \Aut (N)$ acting properly discontinuously and cocompactly on N. This situation is a natural generalization of the so-called affine crystallographic groups. We prove that for all dimensions $1\le n\le 5$ the generalized Auslander conjecture holds, i.e., that such subgroups are virtually polycyclic.

math.DG↗