Neeb / Pianzola Developments and Trends in Infinite-Dimensional Lie Theory

2011
ISBN: 978-0-8176-4741-4
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 288, 492 Seiten

Reihe: Progress in Mathematics

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. Beltita, 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.

Autoren/Hrsg.

Neeb, Karl-Hermann

Pianzola, Arturo

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;Part A Infinite-Dimensional Lie (Super-)Algebras;10
3.1;Isotopy for Extended Affine Lie Algebras and Lie Tori;11
3.1.1;1 Introduction;11
3.1.2;2 Extended affine Lie algebras;13
3.1.3;3 Lie tori;14
3.1.4;4 The construction of EALAs from Lie tori;18
3.1.5;5 Isotopy for Lie tori;21
3.1.6;6 Isotopy in the theory of EALAs;22
3.1.7;7 Coordinatization of Lie tori;30
3.1.8;8 Type A1;32
3.1.9;9 Type A2;36
3.1.10;10 Type Ar, r = 3;40
3.1.11;11 Type Cr, r = 4;42
3.1.12;12 Some concluding remarks;48
3.1.13;References;49
3.2;Remarks on the Isotriviality of Multiloop Algebras;52
3.2.1;1 Introduction;52
3.2.1.1;1.1 Notation and conventions;53
3.2.1.2;1.2 Rn-torsors under finite constant groups;54
3.2.2;2 Bounds on the isotriviality of multiloop algebras;55
3.2.3;References;58
3.3;Extended Affine Lie Algebras and Other Generalizations of Affine Lie Algebras – A Survey;59
3.3.1;1 Introduction;59
3.3.2;2 Root systems and other types of reflection systems;64
3.3.3;3 Affine reflection systems;74
3.3.4;4 Graded algebras;82
3.3.5;5 Lie algebras graded by root systems;94
3.3.6;6 Extended affine Lie algebras and generalizations;102
3.3.7;7 Example: slI(A) for A associative;118
3.3.8;References;128
3.4;Tensor Representations of Classical Locally Finite Lie Algebras;133
3.4.1;Introduction;133
3.4.2;1 Preliminaries;134
3.4.3;2 Tensor representations of gl8 and sl8;136
3.4.4;3 Tensor representations of sp8;143
3.4.5;4 Tensor representations of so8;146
3.4.6;5 Tensor representations of root-reductive Lie algebras;150
3.4.7;6 Appendix;151
3.4.8;References;155
3.5;Lie Algebras, Vertex Algebras, and Automorphic Forms;157
3.5.1;1 Introduction;157
3.5.2;2 Generalized Kac–Moody algebras;158
3.5.3;3 Vertex algebras;161
3.5.4;4 Automorphic forms on orthogonal groups;163
3.5.5;5 Moonshine for Conway’s group;166
3.5.6;7 Construction as strings;171
3.5.7;8 Open problems;172
3.5.8;References;173
3.6;Kac–Moody Superalgebras and Integrability;175
3.6.1;1 Introduction;175
3.6.2;2 Basic definitions;177
3.6.3;3 Odd reflections, Weyl groupoid, and principal roots;180
3.6.4;4 Kac–Moody superalgebras;185
3.6.5;5 Regular Kac–Moody superalgebras with two simple roots;188
3.6.6;6 Examples of regular quasisimple superalgebras;188
3.6.7;7 Classification results;194
3.6.8;8 Applications of classification results;195
3.6.9;9 Description of g (B) and g' in examples;197
3.6.10;10 Integrable modules and highest weight modules;200
3.6.11;11 General properties of category O;205
3.6.12;12 Lie superalgebras sl (1|n)(1), osp(2|2n)(1), andS (1, 2; b);208
3.6.13;13 Lie superalgebras sl (1|n)(1), osp(2|2n)(1);214
3.6.14;14 On affine character formulae;217
3.6.15;References;224
4;Part B Geometry of Infinite-Dimensional Lie (Transformation) Groups;225
4.1;Jordan Structures and Non-Associative Geometry;226
4.1.1;Introduction;226
4.1.2;1 Jordan pairs and graded Lie algebras;229
4.1.2.1;1.1 Z/(2)-graded Lie algebras and Lie triple systems;229
4.1.2.2;1.2 3-graded Lie algebras and Jordan pairs;230
4.1.2.3;1.3 Involutive Z-graded Lie algebras;231
4.1.2.4;1.4 The link with Jordan algebras;232
4.1.2.5;1.5 Some examples;233
4.1.3;2 The generalized projective geometry of a Jordan pair;234
4.1.3.1;2.1 The construction;234
4.1.3.2;2.2 Generalized projective geometries;235
4.1.3.3;2.3 The geometric Jordan–Lie functor;237
4.1.3.4;2.4 Examples revisited;238
4.1.4;3 The universal model;240
4.1.4.1;3.1 Ordinary flag geometries;240
4.1.4.2;3.2 Filtrations and gradings of Lie algebras;241
4.1.5;4 The geometry of states;242
4.1.5.1;4.1 Intrinsic subspaces;242
4.1.5.2;4.2 Examples;244
4.1.5.3;References;245
4.2;Direct Limits of Infinite-Dimensional Lie Groups;247
4.2.1;1 Introduction;247
4.2.1.1;1.1 Direct limit properties of ascending unions;249
4.2.1.2;1.2 Existence of direct limit charts: an essential hypothesis;249
4.2.1.3;1.3 Homotopy groups of ascending unions of Lie groups;250
4.2.1.4;1.4 Regularity in Milnor’s sense;251
4.2.1.5;1.5 Subgroups of direct limit groups;252
4.2.1.6;1.6 Constructions of Lie group structures on ascending unions;252
4.2.1.7;1.7 Properties of locally convex direct limits;252
4.2.1.8;1.8 Further comments, and some historical remarks;253
4.2.2;2 Preliminaries, terminology and basic facts;255
4.2.3;3 Direct limits of topological groups;259
4.2.4;4 Non-linear mappings on locally convex direct limits;262
4.2.5;5 Lie group structures on directed unions of Lie groups;266
4.2.6;6 Examples of directed unions of Lie groups;269
4.2.7;7 Direct limit properties of ascending unions;272
4.2.8;8 Regularity in Milnor’s sense;273
4.2.9;9 Homotopy groups of ascending unions of Lie groups;276
4.2.10;References;279
4.3;Lie Groups of Bundle Automorphisms and Their Extensions;285
4.3.1;Introduction;285
4.3.1.1;Notation and basic concepts;289
4.3.2;1 Lie group structures on mapping groups and automorphism groups of bundles;290
4.3.2.1;1.1 Automorphism groups of bundles;290
4.3.2.2;1.2 Mapping groups on non-compact manifolds;292
4.3.3;2 Central extensions of mapping groups;293
4.3.3.1;2.1 Central extensions of C8(M, t);294
4.3.3.2;2.2 Covariance of the Lie algebra cocycles;298
4.3.3.3;2.3 Corresponding Lie group extensions;301
4.3.4;3 Twists and the cohomology of vector fields;305
4.3.4.1;3.1 Some cohomology of the Lie algebra of vector fields;306
4.3.4.2;3.2 Abelian extensions of diffeomorphism groups;310
4.3.5;4 Central extensions of gauge groups;318
4.3.5.1;4.1 Central extensions of gau(P);318
4.3.5.2;4.2 Covariance of the Lie algebra cocycles;321
4.3.5.3;4.3 Corresponding Lie group extensions;324
4.3.6;5 Multiloop algebras;327
4.3.6.1;5.1 The algebraic picture;327
4.3.6.2;5.2 Geometric realization of multiloop algebras;328
4.3.6.3;5.3 A generalization of multiloop algebras;329
4.3.6.4;5.4 Connections to forms of Lie algebras over rings;330
4.3.7;6 Concluding remarks;331
4.3.8;7 Appendix A. Abelian extensions of Lie groups;332
4.3.9;8 Appendix B. Abelian extensions of semidirect sums;335
4.3.10;9 Appendix C. Triviality of the group action on Lie algebra cohomology;338
4.3.11;References;339
4.4;Gerbes and Lie Groups;343
4.4.1;Introduction;343
4.4.2;1 Bundle gerbes;344
4.4.3;2 Connections on bundle gerbes and holonomy;348
4.4.4;3 Bundle gerbes over compact Lie groups;353
4.4.5;4 Structure on loop spaces from bundle gerbes;359
4.4.6;5 Algebraic structures for gerbes;361
4.4.6.1;5.1 Bundle gerbe modules;361
4.4.6.2;5.2 Bundle Gerbe bimodules;362
4.4.7;6 Applications to conformal field theory;365
4.4.8;7 Open questions;366
4.4.9;References;367
5;Part C Representation Theory of Infinite-Dimensional Lie Groups;369
5.1;Functional Analytic Background for a Theory of Infinite-Dimensional Reductive Lie Groups;370
5.1.1;1 Introduction: What a reductive Lie group is supposed to be;370
5.1.2;2 Triangular integrals and factorizations;372
5.1.3;3 Invariant means on groups;378
5.1.4;4 Lifting group decompositions to covering groups;383
5.1.5;5 What a reductive Banach–Lie group could be;387
5.1.6;References;393
5.2;Heat Kernel Measures and Critical Limits;396
5.2.1;1 Introduction;396
5.2.2;2 General constructions;400
5.2.2.1;2.1 Abstract Wiener spaces and Gaussian measures;400
5.2.2.2;2.2 Abstract Wiener groups and heat kernel measures;402
5.2.3;3 Invariance questions;404
5.2.4;4 The Example of Map(X, F) (see [M]);405
5.2.5;5 The critical Sobolev exponent and X = S1;408
5.2.6;6 2D quantum field theory;414
5.2.7;References;417
5.3;Coadjoint Orbits and the Beginnings of a Geometric Representation Theory;419
5.3.1;1 Introduction;419
5.3.2;2 Banach Poisson manifolds;421
5.3.2.1;2.1 The definition;421
5.3.2.2;2.2 Banach symplectic manifolds;422
5.3.3;3 Banach–Lie–Poisson spaces;423
5.3.3.1;3.1 Characterization;423
5.3.3.2;3.2 Examples;424
5.3.4;4 Symplectic leaves;426
5.3.4.1;4.1 Attempt at constructing symplectic leaves;427
5.3.4.2;4.2 Coadjoint orbits in Banach–Lie–Poisson spaces;429
5.3.5;5 Coadjoint orbits in operator spaces;431
5.3.5.1;5.1 Symplectic leaves in preduals of W*-algebras;432
5.3.5.2;5.2 Symplectic leaves in C*-algebras;434
5.3.5.3;5.3 Symplectic leaves in preduals of operator ideals;435
5.3.5.4;5.4 The restricted unitary algebra and its central extension;437
5.3.5.5;5.5 The restricted Grassmannian;443
5.3.5.6;5.6 Hilbert–Schmidt skew-Hermitian operators;444
5.3.6;6 Geometric representation theory;446
5.3.6.1;6.1 GNS unital *-representation;446
5.3.6.2;6.2 The fundamental construction;447
5.3.6.3;6.3 Reproducing kernels;448
5.3.6.4;6.4 Reproducing kernels and GNS-representations;449
5.3.6.5;6.5 Example;451
5.3.6.6;References;453
5.4;Infinite-Dimensional Multiplicity-Free Spaces I: Limits of Compact Commutative Spaces;460
5.4.1;1 Introduction;460
5.4.2;2 Direct limit groups and representations;462
5.4.3;3 Limit theorem for symmetric spaces;463
5.4.4;4 Gelfand pairs and defining representations;469
5.4.5;5 Function algebras;471
5.4.6;6 Pairs related to spheres and Grassmann manifolds;474
5.4.7;7 Limits related to spheres and Grassmann manifolds;477
5.4.8;8 Conclusions;480
5.4.9;References;482
6;Index;483