Gradings of non-graded Hamiltonian Lie algebras

A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras H(2\colon\n;\omega_2) (of dimension one less than a power of $p$) from which we construct infinite-dimensional thin Lie algebras. In the process we provide an explicit identification of H(2\colon\n;\omega_2) with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic).Comment: 36 pages, to be published in J. Austral. Math. Soc. Ser.

Bounded cohomology of discrete groups

Bounded cohomology of groups was first defined by Johnson and Trauber during the seventies in the context of Banach algebras. As an independent and very active research field, however, bounded cohomology started to develop in 1982, thanks to the pioneering paper "Volume and Bounded Cohomology" by M. Gromov, where the definition of bounded cohomology was extended to deal also with topological spaces. The aim of this monograph is to provide an introduction to bounded cohomology of discrete groups and of topological spaces. We also describe some applications of the theory to related active research fields (that have been chosen according to the taste and the knowledge of the author). The book is essentially self-contained. Even if a few statements do not appear elsewhere and some proofs are slighlty different from the ones already available in the literature, the monograph does not contain original results. In the first part of the book we settle the fundamental definitions of the theory, and we prove some (by now classical) results on low-dimensional bounded cohomology and on bounded cohomology of topological spaces. Then we describe how bounded cohomology has proved useful in the study of the simplicial volume of manifolds, for the classification of circle actions, for the definition and the description of maximal representations of surface groups, and in the study of higher rank flat vector bundles (also in relation with the Chern conjecture).Comment: 220 pages, 2 figure

Cohomology of Finite Groups: Interactions and Applications

The cohomology of finite groups is an important tool in many subjects including representation theory and algebraic topology. This meeting was the fourth in a series that has emphasized the interactions of group cohomology with other areas

Koklassentheorie für nilpotente assoziative Algebren

Coclass theory for finite p-groups was introduced by Leedham-Green & Newman in 1980 and has become a very fruitful approach in the investigation of finite p-groups yielding many interesting results. The central aim of this thesis is to initiate a coclass theory for nilpotent associative algebras over fields and hence to gain further insight into their structure. An essential tool in our investigation are the coclass graphs associated to the nilpotent associative F-algebras of a fixed coclass r. We consider several important features of these graphs. We give a complete structure description of the inverse limits associated to the infinite paths in the coclass graphs. Using this structure description we prove that the number of equivalence classes of infinite paths in a coclass graph is finite if and only if r is less or equal than one or the underlying field is finite. A coclass graph is the disjoint union of its maximal descendant trees. We prove that the root of a maximal descendant tree in a coclass graph for coclass r has dimension bounded by 2r. Each maximal descendant tree can contain either zero, one or several infinite paths starting at its root. A coclass tree is defined to contain exactly one infinite path starting at its root. It is shown that each maximal coclass tree has bounded depth, i.e. there is a non-negative integer c, such that for every algebra in the maximal coclass tree the distance to an algebra on the unique infinite path is at most c. We prove that for finite fields a coclass graph consists of finitely many maximal coclass trees and finitely many other vertices. This made coclass graphs over finite fields accessible for computational experiments using the algorithms we present. The striking observation in the experimental data is that all coclass trees in our examples exhibit a periodic pattern. Based on this observation we conjecture that for a fixed finite field and a fixed non-negative integer r all algebras in the associated coclass graphs can be described by finitely many parametrized presentations. We prove this periodicity conjecture for r=0 and r=1.Koklassentheorie für endliche p-Gruppen wurde 1980 von Leedham-Green & Newman eingeführt. Sie ist seitdem zu einem sehr erfolgreichen Ansatz in der Untersuchung von endlichen p-Gruppen geworden. Das Hauptziel dieser Arbeit ist es, eine Koklassentheorie für nilpotente assoziative Algebren über Körpern zu entwickeln und somit weitere Einblicke in deren Struktur zu bekommen. Ein wichtiges Werkzeug sind die Koklassengraphen, die zu den assoziativen F-Algebren einer festen Koklasse r gehören. Wir betrachten verschiedene wichtige Eigenschaften dieser Graphen. Außerdem geben wir eine vollständige Strukturbeschreibung für die inversen Limites, die zu unendlichen Pfaden in den Koklassengraphen gehören. Darauf aufbauend beweisen wir, dass die Anzahl der Äquivalenzklassen von unendlichen Pfaden in einem Koklassengraphen endlich ist, genau dann wenn r kleiner oder gleich 1 ist oder der Körper endlich ist. Ein Koklassengraph ist die disjunkte Vereinigung seiner maximalen Nachfolgerbäume. Wir beweisen, dass die Wurzel eines maximalen Nachfolgerbaums in einem Koklassengraphen für Koklasse r maximal Dimension 2r hat. Die maximalen Nachfolgerbäume können entweder keinen, einen oder mehrere unendliche, an den Wurzeln der Bäume beginnende, Pfade enthalten. Ein Koklassenbaum ist so definiert, dass er genau einen unendlichen, an der Wurzel beginnenden, Pfad enthält. Es wird gezeigt, dass jeder maximale Koklassenbaum beschränkte Tiefe hat, d.h., es gibt eine nichtnegative Zahl c, so dass für jede Algebra in dem maximalen Koklassenbaum der Abstand zu einer Algebra auf dem eindeutigen unendlichen Pfad höchstens c ist. Wir zeigen, dass ein Koklassengraph über einem endlichen Körper aus endlich vielen maximalen Koklassenbäumen und endlich vielen anderen Knoten besteht. Das macht Koklassengraphen über endlichen Körpern zugänglich für Computerexperimente mit Hilfe der in der Arbeit entwickelten Algorithmen. In den experimentellen Daten lässt sich bemerkenswerterweise in allen berechneten Koklassenbäumen ein periodisches Verhalten erkennen. Auf Basis dieser Beobachtung stellen wir die Vermutung auf, dass für einen fest gewählten, endlichen Körper und feste Koklasse r alle Algebren im zugehörigen Koklassengraphen durch endlich viele parametrisierte Präsentationen beschrieben werden können. Wir beweisen diese Periodizitätsvermutung für r=0 und r=1

Zu den Charaktergraden und Automorphismengruppen von endlichen p-Gruppen fester Koklasse

In 1980 Leedham-Green & Newman initialized the coclass theory, which has provided significant new insights into finite p-groups. For a finite p-group G of order p up to the power n and nilpotency class c the coclass of G is defined as n-c. The classification of finite p-groups by their coclass is a challenging and fascinating open problem. A very important step has been made by Eick & Leedham-Green in 2007. They introduced coclass families that are infinite families of finite p-groups of a fixed coclass with many interesting properties. They can be used to classify the 2-groups of fixed coclass and the 3-groups of coclass 1. By giving a counterexample we show that there is a mistake in their paper. We fill this gap here and thus complete their work. By definition all groups of a coclass family can be described by a single parametrized presentation. We extend this to the automorphism groups and the numbers of irreducible characters of the groups of a coclass family. We show in particular that for each coclass family there exists a 2-cocyle and a pro-p-group yielding naturally group presentations of almost all corresponding automorphism groups, and there exists a single polynomial describing the number of irreducible characters of almost all groups in the coclass family.Leedham-Green und Newman stellten 1980 die sogenannten Koklassenvermutungen auf, welche den Ausgangspunkt der Koklassentheorie bildeten. Mit Hilfe der Koklassentheorie konnten viele neue Einsichten in endliche p-Gruppen gewonnen werden. Die Koklasse einer endlichen p-Gruppe G der Ordnung p hoch n und der nilpotenten Länge c ist definiert als n-c. Die Klassifikation endlicher p-Gruppen anhand ihrer Koklasse ist ein forderndes und zugleich faszinierendes offenes Problem. Einen wichtigen Schritt in Richtung einer möglichen Klassifikation stellt die Veröffentlichung von Eick & Leedham-Green dar. Sie führten Koklassenfamilien ein, welche unendliche Familien endlicher p-Gruppen gleicher Koklasse sind. Mithilfe von Koklassenfamilien kann man die endlichen 3-Gruppen der Koklasse 1 und die 2-Gruppen fester Koklasse r klassifizieren. Die Publikation von Eick & Leedham-Green enthält eine fehlerhafte Aussage. Wir zeigen dies, indem wir ein Gegenbeispiel angeben. Ferner beheben wir den Fehler und vervollständigen somit ihre Arbeit. Per definitionem können alle Gruppen einer Koklassenfamilie mit einer einzigen parametrisierten Gruppenpräsentation angegeben werden. In der vorliegenden Arbeit zeigen wir, dass analoge Aussagen für die Automorphismengruppen und die Anzahl der irreduziblen Charaktere der Gruppen einer Koklassenfamilie gelten. Insbesondere zeigen wir, dass für jede Koklassenfamilie ein 2-Kozykel und eine unendliche pro-p-Gruppe existieren, aus denen sich leicht Gruppenpräsentationen der Automorphismengruppen ableiten lassen. Desweiteren gibt es für jede Koklassenfamilie ein Polynom, welche für fast alle Gruppen der Familie die Anzahl der irreduziblen Charaktere beschreibt

Non-Associative Algebraic Structures: Classification and Structure

These are detailed notes for a lecture on "Non-associative Algebraic Structures: Classification and Structure" which I presented as a part of my Agrega\c{c}\~ao em Matem\'atica e Applica\c{c}\~oes (University of Beira Interior, Covilh\~a, Portugal, 13-14/03/2023)

Integrated modelling for 3D GIS

A three dimensional (3D) model facilitates the study of the real world objects it represents. A geoinformation system (GIS) should exploit the 3D model in a digital form as a basis for answering questions pertaining to aspects of the real world. With respect to the earth sciences, different kinds of objects of reality can be realized. These objects are components of the reality under study. At the present state-of-the-art different realizations are usually situated in separate systems or subsystems. This separation results in redundancy and uncertainty when different components sharing some common aspects are combined. Relationships between different kinds of objects, or between components of an object, cannot be represented adequately. This thesis aims at the integration of those components sharing some common aspects in one 3D model. This integration brings related components together, minimizes redundancy and uncertainty. Since the model should permit not only the representation of known aspects of reality, but also the derivation of information from the existing representation, the design of the model is constrained so as to afford these capabilities. The tessellation of space by the network of simplest geometry, the simplicial network, is proposed as a solution. The known aspects of the reality can be embedded in the simplicial network without degrading their quality. The model provides finite spatial units useful for the representation of objects. Relationships between objects can also be expressed through components of these spatial units which at the same time facilitate various computations and the derivation of information implicitly available in the model. Since the simplicial network is based on concepts in geoinformation science and in mathematics, its design can be generalized for n-dimensions. The networks of different dimension are said to be compatible, which enables the incorporation of a simplicial network of a lower dimension into another simplicial network of a higher dimension.The complexity of the 3D model fulfilling the requirements listed calls for a suitable construction method. The thesis presents a simple way to construct the model. The raster technique is used for the formation of the simplicial network embedding the representation of the known aspects of reality as constraints. The prototype implementation in a software package, ISNAP, demonstrates the simplicial network's construction and use. The simplicial network can facilitate spatial and non spatial queries, computations, and 2D and 3D visualizations. The experimental tests using different kinds of data sets show that the simplicial network can be used to represent real world objects in different dimensionalities. Operations traditionally requiring different systems and spatial models can be carried out in one system using one model as a basis. This possibility makes the GIS more powerful and easy to use