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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.
Table des matières
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.
Résumé
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.
