## Infinite-dimensional Lie Groups |

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### Contents

1 | |

5 | |

7 | |

9 | |

4 Rings and groups of linear mappings | 14 |

5 Definition of differentiable mappings | 19 |

6 Implicit function theorems | 24 |

7 Ordinary differential equations Existence and regularity | 27 |

2 Spectra of compact operators | 161 |

3 Spectra of HilbertSchmidt operators | 164 |

4 Adjoint actions and the HilleYoshida theorem | 167 |

5 Elliptic differential operators | 170 |

Normed Lie algebras | 178 |

Several Subgroups of TM | 181 |

2 Multivalued volume forms | 185 |

3 Symplectic transformation groups | 187 |

8 Examples of Sobolev chains | 31 |

InfiniteDimensional Manifolds | 35 |

2 Vector bundles and affine connections | 40 |

3 Covariant exterior derivatives and Lie derivatives | 43 |

4 Bmanifolds and gauge bundles | 46 |

5 Frobenius theorems | 50 |

6 ILHmanifolds and conformal structures | 55 |

7 Groups of bounded operators and Grassmann manifolds | 58 |

Chapter HI InfiniteDimensional Lie Groups | 63 |

2 Finitedimensional subgroups finitecodimensional subgroups | 68 |

3 Strong ILBLie groups | 73 |

4 Lie algebras exponential mappings subgroups | 78 |

5 Strong ILBLie groups are regular FLie groups | 83 |

Geometrical Structures on Orbits | 91 |

2 Geometrical structures defined by Lie algebras | 95 |

3 Structures given by elliptic complexes | 99 |

4 Several remarks | 105 |

Fundamental Theorems for Differentiability | 111 |

2 Bilateral ILBchains and formal adjoint operators | 116 |

3 Differentiability and linear estimates | 120 |

4 Linear mappings of FE into rSirEE | 122 |

5 Differentiability of compositions | 125 |

6 Continuity of the inverse | 127 |

Groups of C Diffeomorphisms on Compact Manifolds | 133 |

2 Groups of diffeomorphisms on compact manifolds | 136 |

3 Several subgroups of VM | 140 |

4 Subgroups of TM leaving a subset S invariant | 142 |

5 Remarks on global hypoellipticity | 146 |

6 Actions on differential forms | 149 |

7 Conjugacy of compact subgroups | 154 |

Linear Operators | 159 |

4 Hamiltonian systems | 191 |

5 Contact algebras and Poisson algebras | 195 |

6 Contact transformations | 198 |

7 Deformation of a regular contact structure | 201 |

Smooth Extension Theorems | 207 |

2 Subbundles defined by invariant bundle homomorphisms | 211 |

3 The Frobenius theorem on strong ILBLie groups | 215 |

4 Elementary smooth extension theorems on DM | 217 |

5 A Smooth extension theorem for differential operators | 220 |

6 The Frobenius theorem for finite codimensional Lie subalgebras | 224 |

7 The implicit function theorem via Frobenius theorem | 226 |

8 Existence of invariant connections and regularity of the exponential mapping | 229 |

Group of Diffeomorphisms on Cotangent Bundles | 233 |

2 Strong ILHLie group with the Lie algebra ETjS11 | 238 |

3 Infinitedimensional Lie groups with Lie algebra S1I | 240 |

4 Regular FLie group with the Lie algebra ETjJ | 244 |

5 Groups of paths and loops | 247 |

6 Extensions by 2cocycles | 251 |

Pseudodifferential Operators on Manifolds | 255 |

Lie Algebra of Vector Fields | 277 |

Quantizations | 293 |

Poisson Manifolds and Quantum Groups | 319 |

Weyl Manifolds | 341 |

InfiniteDimensional Poisson Manifolds | 361 |

Appendix I | 381 |

Appendix II | 389 |

Appendix III | 395 |

403 | |

409 | |

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### Common terms and phrases

affine connection assume B-space bundle homomorphism C N(d called compact computation consider construct contact algebra continuous linear mapping converges coordinate system Corollary defined definition deformation quantization denote derivative diffeomorphism differential equation differential operator easy element exponential mapping extends F-manifold F(TM fiber finite Fourier FS(TM geodesic given group modeled H-space Hamiltonian Hence identity ILB-chain ILB-normal mapping implicit function theorem inequality infinitesimal inner product integral curve integral submanifold invariant Lemma Let G Lie algebra Lie group Lie subalgebra linear isomorphism neighborhood norm notation Note obtain open subset operator of order Poisson algebra polynomial positive constant product integral proof of Theorem Proposition prove pseudodifferential operators regular F-Lie group resp respect Riemannian metric Sobolev manifold strong ILB-Lie group structure subbundle subgroup Suppose symplectic tangent bundle topology transformations trivial vector bundle vector field