## An Introduction to Lie Groups and Lie AlgebrasThis classic graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semisimple Lie algebras. Lie theory, in its own right, has become regarded as a classical branch of mathematics. Written in an informal style, this is a contemporary introduction to the subject which emphasizes the main concepts of the proofs and outlines the necessary technical details, allowing the material to be conveyed concisely. Based on a lecture course given by the author at the State University of New York at Stony Brook, the book includes numerous exercises and worked examples and is ideal for graduate courses on Lie groups and Lie algebras. |

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### Contents

2 | 5 |

Lie groups and Lie algebras | 25 |

1 | 44 |

Representations of Lie groups and Lie algebras | 52 |

Structure theory of Lie algebras | 84 |

12 | 89 |

Complex semisimple Lie algebras | 108 |

Root systems | 132 |

Representations of semisimple Lie algebras | 163 |

Overview of the literature | 197 |

Appendix B Sample syllabus | 210 |

216 | |

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### Common terms and phrases

action of G algebra g basis bilinear form Cartan subalgebra closed Lie subgroup commutator completely reducible complex Lie group complex representation consider Corollary corresponding defined denoted diagonal dimensional direct sum Dynkin diagram eigenspaces eigenvalues Example Exercise exp(x finite finite-dimensional representation functions given GL(n gl(V groups and Lie Haar measure highest weight representation hyperplane implies inner product integral intertwining operators invariant irreducible representations isomorphism Killing form Lemma Let G Lie group G Lie subgroup manifold morphism morphism of Lie multiplication nilpotent non-degenerate non-zero orthogonal polynomial proof of Theorem Proposition prove quotient real Lie group real or complex representation of g representation of sl(2,C representation theory result right-invariant Section semisimple Lie algebra Show simple roots solvable submanifold subrepresentation subspace symmetric theory of Lie unique vector fields vector space Verma module Weyl chamber Weyl character formula Weyl group