## The structure of Lie groups |

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### Contents

Topological Groups 1 | xi |

Compact Groups | 14 |

Elementary Structure Theory | 33 |

Copyright | |

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### Common terms and phrases

abelian adjoint representation analytic homomorphism analytic manifold isomorphism analytic map analytic subgroup analytic tangent vector belongs canonical map Cartan subalgebra Chapter clear closed in G closed subgroup commutator compact group compact subgroup completes the proof complex analytic group complex analytic representation connected component contains continuous homomorphism defined denote direct sum endomorphism evenly covered evidently expG finite follows G-module group G Haar integral Hence ideal identity map induces injective integer kernel Lemma Let G Lie algebra homomorphism Lie group linear endomorphism locally compact morphism nilpotent non-trivial normal subgroup open connected neighborhood open set positive real number power series power series function power series map proof of Theorem Proposition 3.1 proved real analytic group representation of G restriction SC(G semidirect product sequence Sf(G simply connected structure subgroup of G subspace suppose surjective tangent vector field Theorem 3.1 topological group toroid unipotent vector group vector space whence