## Categories of Symmetries and Infinite-dimensional GroupsFor mathematicians working in group theory, the study of the many infinite-dimensional groups has been carried out in an individual and non-coherent way. For the first time, these apparently disparate groups have been placed together, in order to construct the `big picture'. This booksuccessfully gives an account of this - and shows how such seemingly dissimilar types such as the various groups of operators on Hilbert spaces, or current groups are shown to belong to a bigger entitity. This is a ground-breaking text will be important reading for advanced undergraduate andgraduate mathematicians. |

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### Contents

Spinor representation | 17 |

Representations of the complex classical categories | 68 |

Fermion Fock space | 100 |

finite dimensional case | 126 |

infinitedimensional case | 157 |

Representations of the group of diffeomorphisms of | 202 |

group | 212 |

The heavy groups | 250 |

g Infinitedimensional classical groups and almostinvariant | 293 |

Some algebraic constructions of measure theory | 342 |

The real classical categories | 373 |

Bosonfermion correspondence | 382 |

Characteristic LivSic function | 388 |

398 | |

415 | |

### Common terms and phrases

2ero assertion automorphisms Bere2in operator bounded operators canonical category GD circle complex consider consists construction converges corresponding cyclic span defined denote diffeomorphisms disc double cosets easily seen elements embedding equal equivalent example Exercise Let Exercise Prove Exercise Show exists finite Fock space formula functions functor fundamental representations graph group Diff group G Hence Hermitian form highest weight Hilbert space Hilbert-Schmidt operator holomorphic Indef infinite-dimensional integral invariant irreducible representation isomorphic kernel Lemma Let G Let H Lie algebra Lie group linear relation matrix maximal isotropic subspace measure Mor(V morphisms multiplication Neretin non-2ero Note O(oo object obtain Ol'shanskii orthogonal Potapov transform projective representation Proof Let Proof of Theorem Proposition representation of G restriction satisfying the condition scalar product self-adjoint semigroup sequence space H spinor representation subgroup subrepresentation subset Suppose topology U(oo unitary operators unitary representation variables Vectc verify Virasoro algebra