## The Geometry of Infinite-Dimensional GroupsThis monograph gives an overview of various classes of infinite-dimensional Lie groups and their applications in Hamiltonian mechanics, fluid dynamics, integrable systems, gauge theory, and complex geometry. The text includes many exercises and open questions. |

### Contents

1 | |

7 | |

Their Geometry Orbits | 47 |

Diﬀeomorphisms of the Circle and the VirasoroBott Group | 67 |

Groups of Diﬀeomorphisms | 88 |

The Group of Pseudodiﬀerential Symbols | 111 |

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### Common terms and phrases

2-cocycle action acts aﬃne algebra g boundary bracket bundle called central extension circle closed coadjoint orbits cocycle commutator compact complex condition conjugacy classes conjugate connections consider constant construction coordinate corresponding curve deﬁned Deﬁnition denote derivative described diﬀeomorphism diﬀerent diﬀerential dimension discussed element elliptic equation equivalent Euler equation Example Exercise exists fact field ﬁnite ﬁrst formula function gauge geodesic given gives group G Hamiltonian hence holomorphic identity inﬁnite-dimensional integral invariant latter Lie algebra Lie group linear linking loop group manifold matrix metric moduli space multiplication natural Note obtain operators pairing particular Poisson Poisson structure polar projective Proof Proposition Prove quotient reduction Remark representation respect restriction root system SDiﬀ(M simple smooth solutions space structure surface symbols symplectic structure tangent Theorem theory transformations values vector ﬁelds Virasoro zero