## 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

algebra of linear algebraically closed apply associative algebra assume automorphism base field basis called canonical Cartan subalgebra Chapter characteristic clear commutative completely reducible condition consequence consider contains corresponding defined definition denote derivation determined diagram dimensional direct easy elements Exercise exists extension field field of characteristic finite finite•dimensional follows form a basis formula given gives Hence highest weight holds homomorphism ideal identity implies induces integer invariant involution irreducible module isomorphism Lemma linear function linear mapping linear transformations Math matrix module Moreover multiplication nilpotent non•zero obtain polynomial positive roots Proof prove radical recall relation relative representation result roots satisfying Show simple Lie algebras skew solvable split submodule subspace suppose symmetric term Theorem theory tion unique vector space weight write zero