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This major volume presents a comprehensive introduction to thestudy of topological transformation groups with respect to topologicalproblems which can be traced back to the qualitative theory ofdifferential equations, and provides a systematic exposition of thefundamental methods and techniques of abstract topologicaldynamics.
The contents can be divided into two parts. The first part is devotedto a broad overview of the topological aspects of the theory ofdynamical systems (including shift systems and geodesic and horocycleflows). Part Two is more specialized and presents in a systematic waythe fundamental techniques and methods for the study of compact minimaflows and their morphisms. It brings together many results which arescattered throughout the literature, and, in addition, many examplesare worked out in detail.
The primary purpose of this book is to bridge the gap between the`beginner' and the specialist in the field of topological dynamics.All proofs are therefore given in detail. The book will, however, alsobe useful to the specialist and each chapter concludes with additionalresults (without proofs) and references to sources and relatedmaterial.
The prerequisites for studying the book are a background in generaltoplogy and (classical and functional) analysis.
For graduates and researchers wishing to have a good, comprehensiveintroduction to topological dynamics, it will also be of greatinterest to specialists. This volume is recommended as a supplementarytext. ... Read more

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This self-contained book serves both as an introduction to profinite groups and as a reference for specialists in some areas of the theory. It contains complete and clear proofs for most results, many of which appear here in book form for the first time. Suitable as a basis for courses.

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This is a book about representing symmetry in quantum mechanics. The book is on a graduate and/or researcher level and it is written with an attempt to be concise, to respect conceptual clarity and mathematical rigor. The basic structures of quantum mechanics are used to identify the automorphism group of quantum mechanics. The main concept of a symmetry action is defined as a group homomorphism from a given group, the group of symmetries, to the automorphism group of quantum mechanics. The structure of symmetry actions is determined under the assumption that the symmetry group is a Lie group. The Galilei invariance is used to illustrate the general theory by giving a systematic presentation of a Galilei invariant elementary particle. A brief description of the Galilei invariant wave equations is also given.

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Focusing mainly on locally compact and compact rings, the text begins by illustrating results from the theory of topological groups, some of which are very recent results published for the first time. Constructions and examples of topological groups, relations between quasi-components and components in topological groups, minimal topological groups and properties of the finest precompact topology on an Abelian group are all covered. The text then goes on to examine various aspects of the theory of topological rings, covering constructions and examples of compact right topological rings, the local structure of locally compact rings, structure of compact rings, the local nilpotence of nilrings satisfying compactness conditions, groups of units of compact rings, bounded locally compact rings, topologically locally finite and locally topologically nilpotent radicals in the class of locally compact rings, rings of endomorphisms of a locally compact module and tensor products of compact rings. Results on totally bounded topological rings are also given and developed e.g. the structure theory for countably compact rings. This book will be invaluable to researchers and mathematicians with a basic grounding in the theory of rings and general topology. ... Read more

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This book presents applications of group analysis ofdifferential equations to various models used in hydrodynamics. Itcontains many new examples of exact solutions to the boundary valueproblems for the Euler and Navier-Stokes equations. Thesesolutions describe vortex structures in an inviscid fluid, Marangoniboundary layers, thermal gravity convection and other interestingeffects. Moreover, the book provides a new method for findingsolutions of nonlinear partial differential equations, which isillustrated by a number of examples, including equations for flows ofa compressible ideal fluid in two and three dimensions. The work isreasonably self-contained and supplemented by examples of directphysical importance.
Audience: This volume will be of interest to postgraduatestudents and researchers whose work involves partial differentialequations, Lie groups, the mathematics of fluids, mathematical physicsor fluid mechanics. ... Read more

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The theory of Kac lagebras and their duality, elaboratedindependently in the seventies by Kac and Vainermann and bythe authors of this book, has nowreached a state ofmaturity which justifies the publication of acomprehensiveand authoritative account in bookform. Further, the topic of"quantum groups" has recently become very fashionable andattractedthe attention of more and more mathematicians andtheoretical physicists. However a good characterization ofquantum groups among Hopf algebras inanalogy to thecharacterization of Lie groups among locally compact groupsis still missing. It is thus very valuable to develop thegeneraltheory as does this book, with emphasis on theanalytical aspects of thesubject instead of the purelyalgebraic ones.While in the Pontrjagin duality theory of locally compactabelian groups a perfect symmetry existsbetween a group andits dual, this is no longer true in the various dualitytheorems of Tannaka, Krein, Stinespring and others dealingwith non-abelian locally compact groups. Kac (1961) andTakesaki (1972)formulated the objective of finding a goodcategory of Hopf algebras, containing the category oflocally compact groups and fulfilling a perfectduality.The category of Kac algebras developed in this bookfullyanswers the original duality problem, while not yetsufficientlynon-unimodular to include quantum groups.This self-contained account of thetheory will be ofinterest to all researchers working in quantumgroups,particularly those interested in the approach by Lie groupsand Lie algebras or by non-commutative geometry, and moregenerally also to those working in C* algebras ortheoretical physics. ... Read more

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Generalising classical concepts of probability theory, theinvestigation of operator (semi)-stable laws as possible limitdistributions of operator-normalized sums of i.i.d. random variable onfinite-dimensional vector space started in 1969. Currently, thistheory is still in progress and promises interesting applications.Parallel to this, similar stability concepts for probabilities ongroups were developed during recent decades. It turns out that theexistence of suitable limit distributions has a strong impact on thestructure of both the normalizing automorphisms and the underlyinggroup. Indeed, investigations in limit laws led to contractablegroups and - at least within the class of connected groups- to homogeneous groups, in particular to groups that aretopologically isomorphic to a vector space. Moreover, it has beenshown that (semi)-stable measures on groups have a vector spacecounterpart and vice versa.
The purpose of this book is to describe the structure of limit lawsand the limit behaviour of normalized i.i.d. random variables ongroups and on finite-dimensional vector spaces from a common point ofview. This will also shed a new light on the classical situation.Chapter 1 provides an introduction to stability problems on vectorspaces. Chapter II is concerned with parallel investigations forhomogeneous groups and in Chapter III the situation beyond homogeneousLie groups is treated. Throughout, emphasis is laid on the descriptionof features common to the group- and vector space situation.
Chapter I can be understood by graduate students with some backgroundknowledge in infinite divisibility. Readers of Chapters II and III areassumed to be familiar with basic techniques from probability theoryon locally compact groups. ... Read more

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A. Figá Talamanca: Random Fourier series on compact groups.- S. Helgason: Representations of semisimple Lie groups.- H. Jacquet: Représentations des groupes linéaires p-adiques.- G.W. Mackey: Infinite-dimensional group representations and their applications. ... Read more

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This book presents global actions of arbitrary Lie groups onlarge classes of generalised functions by using a novel parametricapproach. This new method extends and completes earlier results of theauthor and collaborators, in which global Lie group actions ongeneralised functions were only defined in the case of projectable orfibre-preserving Lie group actions.
The parametric method opens the possibility of dealing with vastlylarger classes of Lie semigroup actions which still transformsolutions into solutions. These Lie semigroups can contain arbitrarynoninvertible smooth mappings. Thus, they cannot be subsemigroups ofLie groups.
Audience: This volume is addressed to graduate students andresearchers involved in solving linear and nonlinear partialdifferential equations, and in particular, in dealing with the Liegroup symmetries of their classical or generalised solutions. ... Read more

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Two contributions on closely related subjects: the theory of linear algebraic groups and invariant theory, by well-known experts in the fields. The book will be very useful as a reference and research guide to graduate students and researchers in mathematics and theoretical physics.

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This book describes a new and original formalism based on mirror symmetries of Lie groups, Lie algebras and Homogeneous spaces. Special systems of mirrors are used for classification purposes and as an instrument for studies of Homogeneous spaces. Tri-symmetric and arbitrary Riemannian Homogeneous spaces can also be researched in this way.

The book should be of particular interest to researchers in Lie Groups, Lie Algebras, Differential Geometry and their applications but it should also prove useful for other postgraduate and advanced graduate students in mathematics.

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This book represents the fruits of the author's many years ofresearch and teaching. The introductory chapter contains the necessarybackground information from algebra, topology, and geometry of realspaces. Chapter 1 presents more specialized information on associativeand nonassociative algebras and on Lie groups and algebras. InChapters 2 through 6 geometric interpretations of all simple Liegroups of classes A_n, B_n, C_n, and D_n aswell as of finite groups of Lie type are given. In Chapters 5 and 6geometric interpretations of quasisimple and r-quasisimple Lie groupsof the same classes are included. In Chapter 7, for the first timeever, geometric interpretations of all simple and quasisimpleLie groups of exceptional classes G₂, F₄, E₆,E₇, and E₈ are given. The role of exercises is played bythe assertions and theorems given without a full proof, but with theindication that they can be proved analogously to already provedtheorems.
Audience: The book will be of interest to graduate students andresearchers in mathematics and physics. ... Read more

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This unique monograph deals with the development of asymptoticmethods of perturbation theory, making wide use of group- theoreticaltechniques. Various assumptions about specific group properties areinvestigated, and are shown to lead to modifications of existingmethods, such as the Bogoliubov averaging method and thePoincaré--Birkhoff normal form, as well as to the formulationof new ones. The development of normalization techniques of Lie groupsis also treated. The wealth of examples demonstrates how these newgroup theoretical techniques can be applied to analyze specificproblems.
This book will be of interest to researchers and graduate students inthe field of pure and applied mathematics, mechanics, physics,engineering, and biosciences.
... Read more

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Recent research has repeatedly led to connections between important rigidity questions and bounded cohomology. However, the latter has remained by and large intractable. This monograph introduces the functorial study of the continuous bounded cohomology for topological groups, with coefficients in Banach modules. The powerful techniques of this more general theory have successfully solved a number of the original problems in bounded cohomology. As applications, one obtains, in particular, rigidity results for actions on the circle, for representations on complex hyperbolic spaces and on Teichmüller spaces. A special effort has been made to provide detailed proofs or references in quite some generality. ... Read more

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The Pontryagin-van Kampen duality theorem and the Bochner theorem on positive-definite functions are known to be true for certain abelian topological groups that are not locally compact. The book sets out to present in a systematic way the existing material. It is based on the original notion of a nuclear group, which includes LCA groups and nuclear locally convex spaces together with their additive subgroups, quotient groups and products. For (metrizable, complete) nuclear groups one obtains analogues of the Pontryagin duality theorem, of the Bochner theorem and of the Lévy-Steinitz theorem on rearrangement of series (an answer to an old question of S. Ulam). The book is written in the language of functional analysis. The methods used are taken mainly from geometry of numbers, geometry of Banach spaces and topological algebra. The reader is expected only to know the basics of functional analysis and abstract harmonic analysis. ... Read more

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The first part of this book on Discrete Subgroups of Lie Groups is written by E. B. Vinberg, V. V. Gorbatsevich, and O. V. Shvartsman. Various types of discrete subgroups of Lie groups arise in the theory of functions of complex variables, arithmetic, geometry, and crystallography. Since the foundation of their general theory in the 50-60s of this century, considerable and in many respects exhaustive results were obtained. This development is reflected in this survey. Both semisimple and general Lie groups are considered. Part II on Cohomologies of Lie Groups and Algebras is written by B. L. Feigin and D. B. Fuchs. It contains different definitions of cohomologies of Lie groups and (both finite-dimensional and some infinite-dimensional) Lie algebras, the main methods of their calculation, and the results of these calculations. The book can be useful as a reference and research guide to graduate students and researchers in different areas of mathematics and theoretical physics. ... Read more