xii Contents
§2. Geometry of manifolds 238
2.1. Fiber bundles 238
2.2. Geometric objects on manifolds 243
2.3. Natural operations on geometric objects 247
2.4. Integration on manifolds 253
§3. Symplectic and Poisson manifolds 256
3.1. Symplectic manifolds 256
3.2. Poisson manifolds 263
3.3. Mathematical model of classical mechanics 264
3.4. Symplectic reduction 265
Appendix III. Lie Groups and Homogeneous Manifolds 269
§1. Lie groups and Lie algebras 269
1.1. Lie groups 269
1.2. Lie algebras 270
1.3. Five definitions of the functor Lie: G g 274
1.4. Universal enveloping algebras 286
§2. Review of the set of Lie algebras 288
2.1. Sources of Lie algebras 288
2.2. The variety of structure constants 291
2.3. Types of Lie algebras 297
§3. Semisimple Lie algebras 298
3.1. Abstract root systems 298
3.2. Lie algebra sl(2, C) 308
3.3. Root system related to (g, h) 310
3.4. Real forms 315
§4. Homogeneous manifolds 318
4.1. G-sets 318
4.2. G-manifolds 323
4.3. Geometric objects on homogeneous manifolds 325
Appendix IV. Elements of Functional Analysis 333
§1. Infinite-dimensional vector spaces 333
1.1. Banach spaces 333
1.2. Operators in Banach spaces 335
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