The Geometry of Infinite-Dimensional Groups

Author: Boris Khesin,Robert Wendt
Publisher: Springer Science & Business Media
ISBN: 3540772634
Category: Mathematics
Page: 304
View: 6257

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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: 7510

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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: 5206

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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:

Infinite Dimensional Groups and Manifolds

Author: Tilmann Wurzbacher
Publisher: Walter de Gruyter
ISBN: 3110200015
Category: Mathematics
Page: 256
View: 9492

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

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: 7263

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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.

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: 5591

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"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.

Representations of Infinite-dimensional Groups

Author: Rais Salmanovich Ismagilov
Publisher: American Mathematical Soc.
ISBN: 9780821897683
Category: Mathematics
Page: 197
View: 3446

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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.

Introduction to Finite and Infinite Dimensional Lie (Super)algebras

Author: Neelacanta Sthanumoorthy
Publisher: Academic Press
ISBN: 012804683X
Category: Mathematics
Page: 512
View: 3701

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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

Differential Analysis in Infinite Dimensional Spaces

Proceedings of an AMS Special Session Held August 8-10, 1983, with Partial Support from the NSERC (Canada)
Author: Kondagunta Sundaresan
Publisher: American Mathematical Soc.
ISBN: 0821850598
Category: Mathematics
Page: 122
View: 7395

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This volume focuses on developments made in the past two decades in the field of differential analysis in infinite dimensional spaces. New techniques such as ultraproducts and ultrapowers have illuminated the relationship between the geometric properties of Banach spaces and the existence of differentiable functions on the spaces. The wide range of topics covered also includes gauge theories, polar subsets, approximation theory, group analysis of partial differential equations, inequalities, and actions on infinite groups. Addressed to both the expert and the advanced graduate student, the book requires a basic knowledge of functional analysis and differential topology.

Infinite Dimensional Kähler Manifolds

Author: Alan Huckleberry,Tilmann Wurzbacher
Publisher: Birkhäuser
ISBN: 3034882270
Category: Mathematics
Page: 375
View: 2842

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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.

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: 360

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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.

The Geometry of Heisenberg Groups

With Applications in Signal Theory, Optics, Quantization, and Field Quantization
Author: Ernst Binz,Sonja Pods
Publisher: American Mathematical Soc.
ISBN: 0821844954
Category: Mathematics
Page: 299
View: 919

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The three-dimensional Heisenberg group, being the simplest non-commutative Lie group, appears prominently in various applications of mathematics. The goal of this book is to present basic geometric and algebraic properties of the Heisenberg group and its relation to other important mathematical structures (the skew field of quaternions, symplectic structures, and representations) and to describe some of its applications. In particular, the authors address such subjects as well as signal analysis and processing, geometric optics, and quantization. In each case, the authors present necessary details of the applied topic being considered. With no prerequisites beyond the standard mathematical curriculum, this book manages to encompass a large variety of topics being easily accessible in its fundamentals. It can be useful to students and researchers working in mathematics and in applied mathematics.

Conformal Field Theory and Topology

Author: Toshitake Kohno
Publisher: American Mathematical Soc.
ISBN: 9780821821305
Category: Mathematics
Page: 172
View: 9570

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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.

The Geometry and Physics of Knots

Author: Michael Francis Atiyah
Publisher: Cambridge University Press
ISBN: 9780521395540
Category: Mathematics
Page: 78
View: 886

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Deals with an area of research that lies at the crossroads of mathematics and physics. The material presented here rests primarily on the pioneering work of Vaughan Jones and Edward Witten relating polynomial invariants of knots to a topological quantum field theory in 2+1 dimensions. Professor Atiyah presents an introduction to Witten's ideas from the mathematical point of view. The book will be essential reading for all geometers and gauge theorists as an exposition of new and interesting ideas in a rapidly developing area.

Lie Groups, Lie Algebras, and Some of Their Applications

Author: Robert Gilmore
Publisher: Courier Corporation
ISBN: 0486131564
Category: Mathematics
Page: 608
View: 9124

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This text introduces upper-level undergraduates to Lie group theory and physical applications. It further illustrates Lie group theory's role in several fields of physics. 1974 edition. Includes 75 figures and 17 tables, exercises and problems.

Infinite-dimensional Lie Groups. General Theory and Main Examples

Author: Helge Glöckner,Karl-Hermann Neeb
Publisher: Springer
ISBN: 9780387094441
Category: Mathematics
Page: 350
View: 2264

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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

Arithmetic and Geometry

Papers Dedicated to I.R. Shafarevich on the Occasion of His Sixtieth Birthday. Volume II: Geometry
Author: Michael Artin,John Tate
Publisher: Springer Science & Business Media
ISBN: 1475792867
Category: Mathematics
Page: 487
View: 9144

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Categories of Symmetries and Infinite-dimensional Groups

Author: Yu. A. Neretin
Publisher: Oxford University Press
ISBN: 9780198511861
Category: Mathematics
Page: 417
View: 1323

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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.

Loop Groups

Author: Andrew Pressley,Graeme Segal
Publisher: Oxford University Press
ISBN: 0198535619
Category: Literary Criticism
Page: 318
View: 6976

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Loop groups are the simplest class of infinite dimensional Lie groups, and have important applications in elementary particle physics. They have recently been studied intensively, and the theory is now well developed, involving ideas from several areas of mathematics - algebra, geometry, analysis, and combinatorics. The mathematics of quantum field theory is an important ingredient. This book gives a complete and self-contained account of what is known about the subject and it is writtenfrom a geometrical and analytical point of view, with quantum field theory very much in mind. The mathematics used in connection with loop groups is interesting and important beyond its immediate applications and the authors have tried to make the book accessible to mathematicians in many fields. The hardback edition was published in December 1986.