The Geometry of Infinite-Dimensional Groups


Author: Boris Khesin,Robert Wendt
Publisher: Springer Science & Business Media
ISBN: 3540772634
Category: Mathematics
Page: 304
View: 4043

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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 and Manifolds


Author: Tilmann Wurzbacher
Publisher: Walter de Gruyter
ISBN: 3110200015
Category: Mathematics
Page: 256
View: 6276

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

Infinite Dimensional Groups with Applications


Author: Victor Kac
Publisher: Springer Science & Business Media
ISBN: 1461211042
Category: Mathematics
Page: 380
View: 3462

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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: 3387

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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:

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: 4949

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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.

Infinite Dimensional Kähler Manifolds


Author: Alan Huckleberry,Tilmann Wurzbacher
Publisher: Birkhäuser
ISBN: 3034882270
Category: Mathematics
Page: 375
View: 7165

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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.

Categories of Symmetries and Infinite-dimensional Groups


Author: Yu. A. Neretin
Publisher: Oxford University Press
ISBN: 9780198511861
Category: Mathematics
Page: 417
View: 423

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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.

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


Author: Marion Jean,Heyer Herbert
Publisher: World Scientific
ISBN: 9814544841
Category:
Page: 408
View: 4626

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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.

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: 5600

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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 Lie Groups


Author: N.A
Publisher: American Mathematical Soc.
ISBN: 9780821889589
Category: Mathematics
Page: 415
View: 2338

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This book develops, from the viewpoint of abstract group theory, a general theory of infinite-dimensional Lie groups involving the implicit function theorem and the Frobenius theorem. Omori treats as infinite-dimensional Lie groups all the real, primitive, infinite transformation groups studied by E. Cartan. The book discusses several noncommutative algebras such as Weyl algebras and algebras of quantum groups and their automorphism groups. The notion of a noncommutative manifold is described, and the deformation quantization of certain algebras is discussed from the viewpoint of Lie algebras. This edition is a revised version of the book of the same title published in Japanese in 1979.

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: 4656

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* First of three independent, self-contained volumes under the general title, "Lie Theory," featuring original results and survey work from renowned mathematicians. * Contains J. C. Jantzen's "Nilpotent Orbits in Representation Theory," and K.-H. Neeb's "Infinite Dimensional Groups and their Representations." * Comprehensive treatments of the relevant geometry of orbits in Lie algebras, or their duals, and the correspondence to representations. * Should benefit graduate students and researchers in mathematics and mathematical physics.

Dynamics of Infinite-dimensional Groups

The Ramsey-Dvoretzky-Milman Phenomenon
Author: Vladimir Pestov
Publisher: American Mathematical Soc.
ISBN: 9780821882962
Category: Mathematics
Page: 192
View: 9348

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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. Thedynamics 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 spacecontinuously 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 anddescriptive 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: 8613

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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

Conformal Field Theory and Topology


Author: Toshitake Kohno
Publisher: American Mathematical Soc.
ISBN: 9780821821305
Category: Mathematics
Page: 172
View: 1070

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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.

From Geometry to Quantum Mechanics

In Honor of Hideki Omori
Author: Yoshiaki Maeda,Peter Michor,Takushiro Ochiai,Akira Yoshioka
Publisher: Springer Science & Business Media
ISBN: 0817645306
Category: Mathematics
Page: 324
View: 2613

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* Invited articles in differential geometry and mathematical physics in honor of Hideki Omori * Focus on recent trends and future directions in symplectic and Poisson geometry, global analysis, Lie group theory, quantizations and noncommutative geometry, as well as applications of PDEs and variational methods to geometry * Will appeal to graduate students in mathematics and quantum mechanics; also a reference

Handbook of the Geometry of Banach Spaces


Author: William B. Johnson,Joram Lindenstrauss
Publisher: Elsevier
ISBN: 9780444513052
Category: Mathematics
Page: 1866
View: 2070

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Encouraged by new perspectives in Banach space theory, the editors present this second volume that opens with an introductory essay that explains the basics of the theory. The rest of the chapters focus on specific directions of Banach space theory or its applications.

Vladimir I. Arnold - Collected Works

Hydrodynamics, Bifurcation Theory, and Algebraic Geometry 1965-1972
Author: Vladimir I. Arnold
Publisher: Springer Science & Business Media
ISBN: 3642310311
Category: Mathematics
Page: 465
View: 553

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Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This second volume of his Collected Works focuses on hydrodynamics, bifurcation theory, and algebraic geometry.

The Geometry and Topology of Coxeter Groups


Author: Michael Davis
Publisher: Princeton University Press
ISBN: 0691131384
Category: Mathematics
Page: 584
View: 7295

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The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "CAT(0) groups." The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincaré Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.

Introduction to Finite and Infinite Dimensional Lie (Super)algebras


Author: Neelacanta Sthanumoorthy
Publisher: Academic Press
ISBN: 012804683X
Category: Mathematics
Page: 512
View: 7380

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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