## Lie AlgebrasDefinitive treatment covers split semi-simple Lie algebras, universal enveloping algebras, classification of irreducible modules, automorphisms, simple Lie algebras over an arbitrary field, and more. Classic handbook for researchers and students; useable in graduate courses or for self-study. Reader should have basic knowledge of Galois theory and the Wedderburn structure theory of associative algebras. |

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### Contents

Basic Concepts | 1 |

Solvable and Nilpotent Lie Algebras | 31 |

Cartans Criterion and Its Consequences | 57 |

Split SemiSimple Lie Algebras | 107 |

Universal Enveloping Algebras | 151 |

The Theorem of AdoIwasawa | 199 |

Classification of Irreducible Modules | 207 |

Characters of the Irreducible Modules | 239 |

Automorphisms | 265 |

Simple Lie Algebras over an Arbitrary Field | 289 |

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### Common terms and phrases

abelian alge algebra 21 algebra of characteristic algebra of linear algebra of type algebraically closed field automorphism base field bilinear form Cartan matrix Cartan subalgebra central simple characteristic roots commutative completely reducible contains corresponding decomposition defined denote derivation algebra determined dimensional direct sum Dynkin diagram exists field of characteristic finite finite•dimensional Lie algebra finite•dimensional vector space following Theorem form a basis formula group of automorphisms Hence highest weight homomorphism ideal implies induces involution irreducible module isomorphism Lemma linear combination linear mapping linear transformations Math monomials morphism multiplication nilpotent Lie algebra non•associative algebra non•degenerate obtain orthogonal polynomial positive roots Proof prove relative representation restricted Lie algebra result satisfying semi•simple Lie algebra Show simple Lie algebras simple system skew solvable submodule subspace suppose symmetric system of roots universal enveloping algebra vector space weight spaces Weyl