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Infinite Dimensional Kähler Manifolds

Infinite Dimensional Kähler Manifolds
Catalogue Information
Field name Details
Dewey Class 516
Title Infinite Dimensional Kähler Manifolds ([EBook]) / edited by Alan Huckleberry, Tilmann Wurzbacher.
Added Personal Name Huckleberry, Alan
Wurzbacher, Tilmann
Other name(s) SpringerLink (Online service)
Publication Basel : Birkhäuser , 2001.
Physical Details XIII, 375 pages : online resource.
Series DMV Seminar Band ; 31
ISBN 9783034882279
Summary Note 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.:
Contents note to Group Actions in Symplectic and Complex Geometry -- I. Finite-dimensional manifolds -- II. Elements of Lie groups and their actions -- III. Manifolds with additional structure -- IV. Symplectic manifolds with symmetry -- V. Kählerian structures on coadjoint orbits of compact groups and associated representations -- Literature -- Infinite-dimensional Groups and their Representations -- I. Calculus in locally convex spaces -- II. Dual spaces of locally convex spaces -- III. Topologies on function spaces -- IV. Representations of infinite-dimensional groups -- V. Generalized coherent state representations -- References -- Borel-Weil Theory for Loop Groups -- I. Compact groups -- II. Loop groups and their central extensions -- III. Root decompositions -- IV. Representations of loop groups -- V. Representations of involutive semigroups -- VI. Borel-Weil theory -- VII. Consequences for general representations -- References -- Coadjoint Representation of Virasoro-type Lie Algebras and Differential Operators on Tensor-densities -- I. Coadjoint representation of Virasoro group and Sturm-Liouville operators; Schwarzian derivative as a 1-cocycle -- II. Projectively invariant version of the Gelfand-Fuchs cocycle and of the Schwarzian derivative -- III. Kirillov’s method of Lie superalgebras -- IV. Invariants of coadjoint representation of the Virasoro group -- V. Extension of the Lie algebra of first order linear differential operators on S1 and matrix analogue of the Sturm-Liouville operator -- VI. Geometrical definition of the Gelfand-Dickey bracket and the relation to the Moyal-Weil star-product -- References -- From Group Actions to Determinant Bundles Using (Heat-kernel) Renormalization Techniques -- I. Renormalization techniques -- II. The first Chern form on a class of hermitian vector bundles -- III. The geometry of gauge orbits -- IV. The geometry of determinant bundles -- V. An example: the action of diffeomorphisms on complex structures -- References -- Fermionic Second Quantization and the Geometry of the Restricted Grassmannian -- I. Fermionic second quantization -- II. Bogoliubov transformations and the Schwinger term -- III. The restricted Grassmannian of a polarized Hilbert space -- IV. The non-equivariant moment map of the restricted Grassmannian -- V. The determinant line bundle on the restricted Grassmannian -- References.
System details note Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site http://dx.doi.org/10.1007/978-3-0348-8227-9
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