Contents xi
3.4. Supergroups 194
§4. Why the orbit method works 194
4.1. Mathematical argument 194
4.2. Physical argument 196
§5. Byproducts and relations to other domains 198
5.1. Moment map 198
5.2. Integrable systems 199
§6. Some open problems and subjects for meditation 201
6.1. Functional dimension 201
6.2. Infinitesimal characters 203
6.3. Multiplicities and geometry 203
6.4. Complementary series 204
6.5. Finite groups 205
6.6. Infinite-dimensional groups 205
Appendix I. Abstract Nonsense 207
§1. Topology 207
1.1. Topological spaces 207
1.2. Metric spaces and metrizable topological spaces 208
§2. Language of categories 211
2.1. Introduction to categories 211
2.2. The use of categories 214
2.3. Application: Homotopy groups 215
§3. Cohomology 216
3.1. Generalities 216
3.2. Group cohomology 217
3.3. Lie algebra cohomology 219
3.4. Cohomology of smooth manifolds 220
Appendix II. Smooth Manifolds 227
§1. Around the definition 227
1.1. Smooth manifolds. Geometric approach 227
1.2. Abstract smooth manifolds. Analytic approach 230
1.3. Complex manifolds 235
1.4. Algebraic approach 236
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