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**Lie algebras.**
*(English)*
Zbl 0121.27504

The theory of Lie algebras is developed with no reference to groups and to algebraic closedness of the ground field (if needed the latter is replaced by requiring a splitting Cartan subalgebra). The mathematical style is elementary and perspicuous. In details one can notice many improvements on prevailing methods, e.g. the exposition of the theorem on complete reducibility of linear representations of semisimple Lie algebras, Levi’s theorem, Ado’s theorem, and the direct construction of semisimple Lie algebras and their irreducible representations. Beside the classical theory and the subjects we mentioned before, the author deals with the character formulas, automorphisms of Lie algebras, and simple Lie algebras over arbitrary fields. The method of dealing with automorphisms is essentially analytical though formulated in terms of algebraic geometry. The results on automorphisms are of course used in the chapter on Lie algebras over arbitrary fields which reflects fairly well the state of our knowledge with respect to the classification of simple Lie algebras.

Reviewer: Hans Freudenthal (Utrecht)

### MSC:

17Bxx | Lie algebras and Lie superalgebras |

17-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras |

17B05 | Structure theory for Lie algebras and superalgebras |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

17B20 | Simple, semisimple, reductive (super)algebras |

17B40 | Automorphisms, derivations, other operators for Lie algebras and super algebras |