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Infinite Dimensional Kahler Manifolds : Oberwolfach Seminars - A. Huckleberry

Infinite Dimensional Kahler Manifolds

Oberwolfach Seminars

By: A. Huckleberry (Editor), T. Wurzbacher (Editor)

Paperback

Published: January 2001
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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 Ka'hler 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.

Preface
Introduction to Group Actions in Symplectic and Complex Geometry
Finite-dimensional manifoldsp. 1
Element of Lie groups and their actionsp. 20
Manifolds with additional structurep. 40
Symplectic manifolds with symmetryp. 82
Kahlerian structures on coadjoint orbits of compact groups and associated representationsp. 95
Infinite-dimensional Groups and their Representations
Calculus in locally convex spacesp. 132
Dual spaces of locally convex spacesp. 144
Topologies on function spacesp. 153
Representations of infinite-dimensional groupsp. 164
Generalized coherent state representationsp. 168
Borel-Weil Theory for Loop Groups
Compact groupsp. 180
Loop groups and their central extensionsp. 181
Root decompositionsp. 197
Representations of loop groupsp. 201
Representations of involutive semigroupsp. 209
Borel-Weil theoryp. 213
Consequences for general representationsp. 226
Coadjoint Representation of Virasoro-type Lie Algebras and Differential Operators on Tensor-densities
Coadjoint representation of Virasoro group and Sturm-Liouville operators; Schwarzian derivative as a 1-cocyclep. 233
Projectively invariant version of the Gelfand-Fuchs cocycle and of the Schwarzian derivativep. 238
Kirillov's method of Lie superalgebrasp. 241
Invariants of coadjoint representation of the Virasoro groupp. 244
Extension of the Lie algebra of first order linear differential operators on S[superscript 1] and matrix analogue of the Sturm-Liouville operatorp. 247
Geometrical definition of the Gelfand-Dickey bracket and the relation to the Moyal-Weil star-productp. 249
From Group Actions to Determinant Bundles Using (Heat-kernel) Renormalization Techniques
Renormalization techniquesp. 260
The first Chern form on a class of hermitian vector bundlesp. 266
The geometry of gauge orbitsp. 270
The geometry of determinant bundlesp. 275
An example: the action of diffeomorphisms on complex structuresp. 278
Fermionic Second Quantization and the Geometry of the Restricted Grassmannian
Fermionic second quantizationp. 290
Bogoliubov transformations and the Schwinger termp. 298
The restricted Grassmannian of a polarized Hilbert spacep. 326
The non-equivariant moment map of the restricted Grassmannianp. 336
The determinant line bundle on the restricted Grassmannianp. 358
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783764366025
ISBN-10: 3764366028
Series: Oberwolfach Seminars
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 375
Published: January 2001
Publisher: Birkhauser Verlag AG
Country of Publication: CH
Dimensions (cm): 24.41 x 16.99  x 2.06
Weight (kg): 0.76