The Geometry of Infinite-Dimensional Groups

Author: Boris Khesin,Robert Wendt

Publisher: Springer Science & Business Media

ISBN: 3540772634

Category: Mathematics

Page: 304

View: 2081

This monograph gives an overview of various classes of infinite-dimensional Lie groups and their applications in Hamiltonian mechanics, fluid dynamics, integrable systems, gauge theory, and complex geometry. The text includes many exercises and open questions.

Infinite Dimensional Groups with Applications

Author: Victor Kac

Publisher: Springer Science & Business Media

ISBN: 1461211042

Category: Mathematics

Page: 380

View: 1254

This volume records most of the talks given at the Conference on Infinite-dimensional Groups held at the Mathematical Sciences Research Institute at Berkeley, California, May 10-May 15, 1984, as a part of the special program on Kac-Moody Lie algebras. The purpose of the conference was to review recent developments of the theory of infinite-dimensional groups and its applications. The present collection concentrates on three very active, interrelated directions of the field: general Kac-Moody groups, gauge groups (especially loop groups) and diffeomorphism groups. I would like to express my thanks to the MSRI for sponsoring the meeting, to Ms. Faye Yeager for excellent typing, to the authors for their manuscripts, and to Springer-Verlag for publishing this volume. V. Kac INFINITE DIMENSIONAL GROUPS WITH APPLICATIONS CONTENTS The Lie Group Structure of M. Adams. T. Ratiu 1 Diffeomorphism Groups and & R. Schmid Invertible Fourier Integral Operators with Applications On Landau-Lifshitz Equation and E. Date 71 Infinite Dimensional Groups Flat Manifolds and Infinite D. S. Freed 83 Dimensional Kahler Geometry Positive-Energy Representations R. Goodman 125 of the Group of Diffeomorphisms of the Circle Instantons and Harmonic Maps M. A. Guest 137 A Coxeter Group Approach to Z. Haddad 157 Schubert Varieties Constructing Groups Associated to V. G. Kac 167 Infinite-Dimensional Lie Algebras I. Kaplansky 217 Harish-Chandra Modules Over the Virasoro Algebra & L. J. Santharoubane 233 Rational Homotopy Theory of Flag S.

Infinite Dimensional Lie Groups in Geometry and Representation Theory

Author: Augustin Banyaga,Joshua A Leslie,Thierry Robart

Publisher: World Scientific

ISBN: 9814488143

Category: Science

Page: 176

View: 9235

This book constitutes the proceedings of the 2000 Howard conference on “Infinite Dimensional Lie Groups in Geometry and Representation Theory”. It presents some important recent developments in this area. It opens with a topological characterization of regular groups, treats among other topics the integrability problem of various infinite dimensional Lie algebras, presents substantial contributions to important subjects in modern geometry, and concludes with interesting applications to representation theory. The book should be a new source of inspiration for advanced graduate students and established researchers in the field of geometry and its applications to mathematical physics. Contents:Inheritance Properties for Lipschitz-Metrizable Frölicher Groups (J Teichmann)Around the Exponential Mapping (T Robart)On a Solution to a Global Inverse Problem with Respect to Certain Generalized Symmetrizable Kac-Moody Algebras (J A Leslie)The Lie Group of Fourier Integral Operators on Open Manifolds (R Schmid)On Some Properties of Leibniz Algebroids (A Wade)On the Geometry of Locally Conformal Symplectic Manifolds (A Banyaga)Some Properties of Locally Conformal Symplectic Manifolds (S Haller)Criticality of Unit Contact Vector Fields (P Rukimbira)Orbifold Homeomorphism and Diffeomorphism Groups (J E Borzellino & V Brunsden)A Note on Isotopies of Symplectic and Poisson Structures (A Banyaga & P Donato)Remarks on Actions on Compacta by Some Infinite-Dimensional Groups (V Pestov) Readership: Graduate students and researchers in mathematics and mathematical physics. Keywords:

Developments and Trends in Infinite-Dimensional Lie Theory

Author: Karl-Hermann Neeb,Arturo Pianzola

Publisher: Springer Science & Business Media

ISBN: 9780817647414

Category: Mathematics

Page: 492

View: 8114

This collection of invited expository articles focuses on recent developments and trends in infinite-dimensional Lie theory, which has become one of the core areas of modern mathematics. The book is divided into three parts: infinite-dimensional Lie (super-)algebras, geometry of infinite-dimensional Lie (transformation) groups, and representation theory of infinite-dimensional Lie groups. Contributors: B. Allison, D. Beltiţă, W. Bertram, J. Faulkner, Ph. Gille, H. Glöckner, K.-H. Neeb, E. Neher, I. Penkov, A. Pianzola, D. Pickrell, T.S. Ratiu, N.R. Scheithauer, C. Schweigert, V. Serganova, K. Styrkas, K. Waldorf, and J.A. Wolf.

Representations of Infinite-dimensional Groups

Author: Rais Salmanovich Ismagilov

Publisher: American Mathematical Soc.

ISBN: 9780821897683

Category: Mathematics

Page: 197

View: 5354

This book is devoted to representations of two classes of infinite-dimensional groups: current groups and diffeomorphism groups. The author presents a complete treatment of the subject, including general methods for constructing irreducible representations of infinite-dimensional groups and general results about such representations. He also exhibits deep relations between representations of infinite-dimensional grops and the theory of Fock spaces, the theory of point random processes, and other branches of mathematics.

Infinite Dimensional Groups and Manifolds

Author: Tilmann Wurzbacher

Publisher: Walter de Gruyter

ISBN: 3110200015

Category: Mathematics

Page: 256

View: 8490

Dieser Band beinhaltet eine Sammlung wissenschaftlicher Forschungsbeiträge zu unendlich-dimensionalen Gruppen und Mannigfaltigkeiten in der Mathematik und Quantenphysik.

Categories of Symmetries and Infinite-dimensional Groups

Author: Yu. A. Neretin

Publisher: Oxford University Press

ISBN: 9780198511861

Category: Mathematics

Page: 417

View: 2242

For mathematicians working in group theory, the study of the many infinite-dimensional groups has been carried out in an individual and non-coherent way. For the first time, these apparently disparate groups have been placed together, in order to construct the `big picture'. This book successfully gives an account of this - and shows how such seemingly dissimilar types such as the various groups of operators on Hilbert spaces, or current groups are shown to belong to a bigger entitity.This is a ground-breaking text will be important reading for advanced undergraduate and graduate mathematicians.

Analysis On Infinite-dimensional Lie Groups And Algebras - Proceedings Of The International Colloquium

Author: Marion Jean,Heyer Herbert

Publisher: World Scientific

ISBN: 9814544841


Page: 408

View: 3820

The book provides a comprehensive “map” of China's financial markets and institutions based on objective data. The book uses the mentioned data to analyze the status and trend of China's financial sectors under macro-economy. The objective of this book is to show the actual performance of China's financial markets and institutions during the first stage of the post-crisis period and the challenges that China's financial sectors face in the future.At present, China's economy and financial sectors are just like a traveler undergoing a long journey and need a map to tell where he/she comes from, where he/she is and where the present road will lead to. This book attempts to provide the readers with some useful information on the basis of objective data and help them to explore the road to the near future of China's economy and financial sectors.

Einführung in die Mechanik und Symmetrie

Eine grundlegende Darstellung klassischer mechanischer Systeme

Author: Jerrold E. Marsden,Tudor S. Ratiu

Publisher: Springer-Verlag

ISBN: 3642568599

Category: Mathematics

Page: 598

View: 2851

Symmetrie spielt in der Mechanik eine große Rolle. Dieses Buch beschreibt die Entwicklung zugrunde liegender Theorien. Besonderes Gewicht wird der Symmetrie beigemessen. Ursache hierfür sind Entwicklungen im Bereich dynamischer Systeme, der Einsatz geometrischer Verfahren und neue Anwendungen. Dieses Lehrbuch stellt Grundlagen bereit und beschreibt zahlreiche spezifische Anwendungen. Interessant für Physiker und Ingenieure. Ausgewählte Beispiele, Anwendungen, aktuelle Verfahren/Techniken veranschaulichen die Theorie.

Einführung in die Geometrie und Topologie

Author: Werner Ballmann

Publisher: Springer-Verlag

ISBN: 3034809018

Category: Mathematics

Page: 162

View: 6217

Das Buch bietet eine Einführung in die Topologie, Differentialtopologie und Differentialgeometrie. Es basiert auf Manuskripten, die in verschiedenen Vorlesungszyklen erprobt wurden. Im ersten Kapitel werden grundlegende Begriffe und Resultate aus der mengentheoretischen Topologie bereitgestellt. Eine Ausnahme hiervon bildet der Jordansche Kurvensatz, der für Polygonzüge bewiesen wird und eine erste Idee davon vermitteln soll, welcher Art tiefere topologische Probleme sind. Im zweiten Kapitel werden Mannigfaltigkeiten und Liesche Gruppen eingeführt und an einer Reihe von Beispielen veranschaulicht. Diskutiert werden auch Tangential- und Vektorraumbündel, Differentiale, Vektorfelder und Liesche Klammern von Vektorfeldern. Weiter vertieft wird diese Diskussion im dritten Kapitel, in dem die de Rhamsche Kohomologie und das orientierte Integral eingeführt und der Brouwersche Fixpunktsatz, der Jordan-Brouwersche Zerlegungssatz und die Integralformel von Stokes bewiesen werden. Das abschließende vierte Kapitel ist den Grundlagen der Differentialgeometrie gewidmet. Entlang der Entwicklungslinien, die die Geometrie der Kurven und Untermannigfaltigkeiten in Euklidischen Räumen durchlaufen hat, werden Zusammenhänge und Krümmung, die zentralen Konzepte der Differentialgeometrie, diskutiert. Den Höhepunkt bilden die Gaussgleichungen, die Version des theorema egregium von Gauss für Untermannigfaltigkeiten beliebiger Dimension und Kodimension. Das Buch richtet sich in erster Linie an Mathematik- und Physikstudenten im zweiten und dritten Studienjahr und ist als Vorlage für ein- oder zweisemestrige Vorlesungen geeignet.

Lie Theory

Lie Algebras and Representations

Author: Jens Carsten Jantzen,Jean-Philippe Anker,Karl-Hermann Neeb,Bent Orsted

Publisher: Springer Science & Business Media

ISBN: 9780817633738

Category: Mathematics

Page: 328

View: 2374

"Lie Theory," a set of three independent, self-contained volumes, features surveys and original work by well-established researchers in key areas of semisimple Lie groups. A wide range of topics is covered, including unitary representation theory and harmonic analysis. "Lie Theory: Lie Algebras and Representations" contains J. C. Jantzen's "Nilpotent Orbits in Representation Theory," and K.-H. Neeb's "Infinite Dimensional Groups and their Representations." Both papers are comprehensive treatments of the relevant geometry of orbits in Lie algebras, or their duals, and the correspondence to representations. Ideal for graduate students and researchers, each volume of "Lie Theory" provides a broad, clearly focused examination of semisimple Lie groups and their integral importance to research in many branches of mathematics.

Dynamics of Infinite-dimensional Groups

The Ramsey-Dvoretzky-Milman Phenomenon

Author: Vladimir Pestov

Publisher: Amer Mathematical Society


Category: Mathematics

Page: 192

View: 9909

The ""infinite-dimensional groups"" in the title refer to unitary groups of Hilbert spaces, the infinite symmetric group, groups of homeomorphisms of manifolds, groups of transformations of measure spaces, etc. The book presents an approach to the study of such groups based on ideas from geometric functional analysis and from exploring the interplay between dynamical properties of those groups, combinatorial Ramsey-type theorems, and the phenomenon of concentration of measure. The dynamics of infinite-dimensional groups is very much unlike that of locally compact groups. For instance, every locally compact group acts freely on a suitable compact space (Veech). By contrast, a 1983 result by Gromov and Milman states that whenever the unitary group of a separable Hilbert space continuously acts on a compact space, it has a common fixed point. In the book, this new fast-growing theory is built strictly from well-understood examples up. The book has no close counterpart and is based on recent research articles. At the same time, it is organized so as to be reasonably self-contained. The topic is essentially interdisciplinary and will be of interest to mathematicians working in geometric functional analysis, topological and ergodic dynamics, Ramsey theory, logic and descriptive set theory, representation theory, topological groups, and operator algebras.

Infinite-dimensional Lie Groups. General Theory and Main Examples

Author: Helge Glöckner,Karl-Hermann Neeb

Publisher: Springer

ISBN: 9780387094441

Category: Mathematics

Page: 350

View: 7915

Provides a comprehensive introduction to this important subject, examining the basic structure theory of infinite-dimensional Lie groups Essentially self-contained, provides all necessary background, excepting modest prerequisites Clear exposition includes careful explanations, illustrative examples, numerous exercises, and detailed cross-references to simplify a non-linear reading of the material

Conformal Field Theory and Topology

Author: Toshitake Kohno

Publisher: American Mathematical Soc.

ISBN: 9780821821305

Category: Mathematics

Page: 172

View: 1718

The aim of this book is to provide the reader with an introduction to conformal field theory and its applications to topology. The author starts with a description of geometric aspects of conformal field theory based on loop groups. By means of the holonomy of conformal field theory he defines topological invariants for knots and 3-manifolds. He also gives a brief treatment of Chern-Simons perturbation theory.

Infinite Dimensional Kähler Manifolds

Author: Alan Huckleberry,Tilmann Wurzbacher

Publisher: Birkhäuser

ISBN: 3034882270

Category: Mathematics

Page: 375

View: 6142

Infinite dimensional manifolds, Lie groups and algebras arise naturally in many areas of mathematics and physics. Having been used mainly as a tool for the study of finite dimensional objects, the emphasis has changed and they are now frequently studied for their own independent interest. On the one hand this is a collection of closely related articles on infinite dimensional Kähler manifolds and associated group actions which grew out of a DMV-Seminar on the same subject. On the other hand it covers significantly more ground than was possible during the seminar in Oberwolfach and is in a certain sense intended as a systematic approach which ranges from the foundations of the subject to recent developments. It should be accessible to doctoral students and as well researchers coming from a wide range of areas. The initial chapters are devoted to a rather selfcontained introduction to group actions on complex and symplectic manifolds and to Borel-Weil theory in finite dimensions. These are followed by a treatment of the basics of infinite dimensional Lie groups, their actions and their representations. Finally, a number of more specialized and advanced topics are discussed, e.g., Borel-Weil theory for loop groups, aspects of the Virasoro algebra, (gauge) group actions and determinant bundles, and second quantization and the geometry of the infinite dimensional Grassmann manifold.

Introduction to Finite and Infinite Dimensional Lie (Super)algebras

Author: Neelacanta Sthanumoorthy

Publisher: Academic Press

ISBN: 012804683X

Category: Mathematics

Page: 512

View: 513

Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. Introduction to Finite and Infinite Dimensional Lie Algebras and Superalgebras introduces the theory of Lie superalgebras, their algebras, and their representations. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semi-simple Lie algebras. While discussing all classes of finite and infinite dimensional Lie algebras and Lie superalgebras in terms of their different classes of root systems, the book focuses on Kac-Moody algebras. With numerous exercises and worked examples, it is ideal for graduate courses on Lie groups and Lie algebras. Discusses the fundamental structure and all root relationships of Lie algebras and Lie superalgebras and their finite and infinite dimensional representation theory Closely describes BKM Lie superalgebras, their different classes of imaginary root systems, their complete classifications, root-supermultiplicities, and related combinatorial identities Includes numerous tables of the properties of individual Lie algebras and Lie superalgebras Focuses on Kac-Moody algebras

Complex Analysis on Infinite Dimensional Spaces

Author: Sean Dineen

Publisher: Springer Science & Business Media

ISBN: 1447108698

Category: Mathematics

Page: 543

View: 6960

Infinite dimensional holomorphy is the study of holomorphic or analytic func tions over complex topological vector spaces. The terms in this description are easily stated and explained and allow the subject to project itself ini tially, and innocently, as a compact theory with well defined boundaries. However, a comprehensive study would include delving into, and interacting with, not only the obvious topics of topology, several complex variables theory and functional analysis but also, differential geometry, Jordan algebras, Lie groups, operator theory, logic, differential equations and fixed point theory. This diversity leads to a dynamic synthesis of ideas and to an appreciation of a remarkable feature of mathematics - its unity. Unity requires synthesis while synthesis leads to unity. It is necessary to stand back every so often, to take an overall look at one's subject and ask "How has it developed over the last ten, twenty, fifty years? Where is it going? What am I doing?" I was asking these questions during the spring of 1993 as I prepared a short course to be given at Universidade Federal do Rio de Janeiro during the following July. The abundance of suit able material made the selection of topics difficult. For some time I hesitated between two very different aspects of infinite dimensional holomorphy, the geometric-algebraic theory associated with bounded symmetric domains and Jordan triple systems and the topological theory which forms the subject of the present book.

Einführung in die Funktionalanalysis

Author: Friedrich Hirzebruch,Winfried Scharlau

Publisher: Spektrum Akademischer Verlag

ISBN: 9783860254295

Category: Mathematics

Page: 178

View: 3131