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Infinite Dimensional Kähler Manifolds: Oberwolfach Seminars, cartea 31

Editat de Alan Huckleberry, Tilmann Wurzbacher
en Limba Engleză Paperback – sep 2001
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.
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Specificații

ISBN-13: 9783764366025
ISBN-10: 3764366028
Pagini: 392
Ilustrații: XIII, 375 p.
Greutate: 0.62 kg
Ediția:2001
Editura: Birkhäuser Basel
Colecția Birkhäuser
Seria Oberwolfach Seminars

Locul publicării:Basel, Switzerland

Public țintă

Research

Descriere

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.

Cuprins

to Group Actions in Symplectic and Complex Geometry.- I. Finite-dimensional manifolds.- 1. Vector space structures.- 2. Local theory.- 3. Global differentiable objects.- 4. A sketch of integration theory.- 5. Smooth submanifolds.- 6. Induced orientation and Stokes’ theorem.- 7. Functionals on de Rham cohomology.- II. Elements of Lie groups and their actions.- 1. Introduction to actions and quotients.- 2. Examples of Lie groups.- 3. Smooth actions of Lie groups.- 4. Fiber bundles.- III. Manifolds with additional structure.- 1. Geometric structures on vector spaces.- 2. The elements of function theory.- 3. A brief introduction to complex analysis in higher dimensions.- 4. Complex manifolds.- 5. Symplectic manifolds.- 6. Kähler manifolds.- IV. Symplectic manifolds with symmetry.- 1. Introduction to the moment map.- 2. Central extensions.- 3. Existence and uniqueness of the moment map.- 4. Basic examples of the moment map.- 5. The Poisson structure on (Lie G)* and on coadjoint orbits.- 6. The basic formula and some consequences.- 7. Moment maps associated to representations.- V. Kählerian structures on coadjoint orbits of compact groups and associated representations.- 1. Generalities on compact groups.- 2. Root decomposition for $${\mathfrak{k}^\mathbb{C}}$$.- 3. Complexification of compact groups.- 4. Algebraicity properties of complexifications of compact groups.- 5. Compact complex homogeneous spaces.- 6. The root groups SL2(?) and H2(G/P, ?).- 7. Representations of complex semisimple groups.- Literature.- Infinite-dimensional Groups and their Representations.- I. Calculus in locally convex spaces.- 1. Differentiable functions.- 2. Differentiable functions on Banach spaces.- 3. Holomorphic functions.- 4. Differentiable manifolds.- 5 Infinite-dimensional Lie groups.- II. Dual spaces of locally convex spaces.- 1. Metrizability.- 2. Semireflexivity.- 3. Completeness properties of the dual space.- III. Topologies on function spaces.- 1. The space C? (M, V).- 2. Smooth mappings between function spaces.- 3. Applications to groups of continuous mappings.- 4. Spaces of holomorphic functions.- IV. Representations of infinite-dimensional groups.- V. Generalized coherent state representations.- 1. The line bundle over the projective space of a topological vector space.- 2. Applications to representation theory.- References.- Borel-Weil Theory for Loop Groups.- I. Compact groups.- II. Loop groups and their central extensions.- 1. Groups of smooth maps.- 2. Central extensions of loop groups.- 3. Appendix IIa: Central extensions and semidirect products.- 4. Appendix IIb: Smoothness of group actions.- 5. Appendix IIc: Lifting automorphisms to central extensions.- 6. Appendix IId: Lifting automorphic group actions to central extensions.- III. Root decompositions.- 1. The Weyl group.- 2. Root decomposition of the central extension.- IV. Representations of loop groups.- 1. Lowest weight vectors and antidominant weights.- 2. The Casimir operator.- 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.- 1. Virasoro group and Virasoro algebra.- 2. Regularized dual space.- 3. Coadjoint representation of the Virasoro algebra.- 4. The coadjoint action of Virasoro group and Schwarzian derivative.- 5. Space of Sturm-Liouville equations as a Diff+(S1)-module.- 6. The isomorphism.- 7. Vect(S1)-action on the space of Sturm-Liouville operators.- II. Projectively invariant version of the Gelfand-Fuchs cocycle and of the Schwarzian derivative.- 1. Modified Gelfand-Fuchs cocycle.- 2. Modified Schwarzian derivative.- 3. Energy shift.- 4. Projective structures.- III. Kirillov’s method of Lie superalgebras.- 1. Lie superalgebras.- 2. Ramond and Neveu-Schwarz superalgebras.- 3. Coadjoint representation.- 4. Projective equivariance and Lie superalgebra osp(1|2).- IV. Invariants of coadjoint representation of the Virasoro group.- 1. Monodromy operator as a conjugation class of $$\widetilde {SL}(2,R)$$.- 2. Classification theorem.- V. Extension of the Lie algebra of first order linear differential operators on S1 and matrix analogue of the Sturm-Liouville operator.- 1. Lie algebra of first order differential operators on S1 and its central extensions.- 2. Matrix Sturm-Liouville operators.- 3. Action of Lie algebra of differential operators.- 4. Generalized Neveu-Schwarz superalgebra.- VI. Geometrical definition of the Gelfand-Dickey bracket and the relation to the Moyal-Weil star-product.- 1. Moyal-Weyl star-product.- 2. Moyal-Weyl star-product on tensor-densities, the transvectants.- 3. Space of third order linear differential operators as a Diff+(S1)-module.- 4. Second order Lie derivative.- 5. Adler-Gelfand-Dickey Poisson structure.- References.- From Group Actions to Determinant Bundles Using (Heat-kernel) Renormalization Techniques.- I. Renormalization techniques.- 1. Renormalized limits.- 2. Renormalization procedures.- 3. Heat-kernel renormalization procedures.- 4. Renormalized determinants.- II. The first Chern form on a class of hermitian vector bundles.- 1. Renormalization procedures on vector bundles.- 2. Weighted first Chern forms on infinite dimensional vector bundles.- III. The geometry of gauge orbits.- 1. The finite dimensional setting.- 2. The infinite dimensional setting.- IV. The geometry of determinant bundles.- 1. Determinant bundles.- 2. A metric on the determinant bundle.- 3. A connection on the determinant bundle.- 4. Curvature on the determinant bundle.- V. An example: the action of diffeomorphisms on complex structures.- 1. The orbit picture.- 2. Riemannian structures.- 3. A super vector bundle arising from the group action.- 4. The determinant bundle picture.- 5. First Chern form on the vector bundle.- References.- Fermionic Second Quantization and the Geometry of the Restricted Grassmannian.- I. Fermionic second quantization.- 1. The Dirac equation and the negative energy problem.- 2. Fermionic multiparticle formalism: Fock space and the CAR-algebra.- II. Bogoliubov transformations and the Schwinger term.- 1. Implementation of operators on the Fock space.- 2. The Schwinger term.- 3. The central extensions Ures~ and GLres~.- III. The restricted Grassmannian of a polarized Hilbert space.- 1. The restricted Grassmannian as a homogeneous complex manifold.- 2. The basic differential geometry of the restricted Grassmannian.- IV. The non-equivariant moment map of the restricted Grassmannian.- 1. Differential k-forms in infinite dimensions.- 2. Symplectic manifolds, group actions and the co-moment map.- 3. Co-momentum and momentum maps (in infinite dimensions).- 4. Examples of symplectic actions and (co-)momementum maps.- 5. The Ures-moment map on Gres and the Schwinger term.- V. The determinant line bundle on the restricted Grassmannian.- 1. The C*-algebraic construction of the determinant bundle DET.- 2. Comparison to other approaches to the determinant bundle.- 3. Holomorphic sections of the dual of DET.- References.