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**Notes on Lie algebras.
2nd. ed.**
*(English)*
Zbl 0708.17005

Universitext. New York etc.: Springer-Verlag. xii, 162 p. DM 58.00 (1990).

The book is the revised edition of the author’s “Notes on Lie Algebras” published by Van Nostrand in 1969 [see the review in Zbl 0209.06601]. The aim of the author is to present the structure and representation theory of semisimple Lie algebras over the complex field in a straightforward and simple manner. The first chapter deals with basic facts; Cartan’s criterion, representations of \(sl(2,\mathbb{C})\) and theorems of Engel and Lie. The structure theory is the topic of the second chapter and the usual players appear. The final chapter is devoted to representation theory (finite dimensional with highest weight vectors, complete reducibility and the Weyl character formula. Use of the PoincarĂ©-Birkhoff-Witt theorem is avoided. Material which did not appear in the first edition but is presented here include the formulas of Freudenthal and Klimyk for the multiplicities of weights, R. Brauer’s algorithm for the splitting of tensor products and the Bose-Patera approach to results of Malcev on which representations of various semisimple Lie algebras can be brought to orthogonal or symplectic form. This text is one which would do well in the classroom after a good linear algebra course or could be used for self-study.

Reviewer: Ernest L. Stitzinger (Raleigh)

### MSC:

17B05 | Structure theory for Lie algebras and superalgebras |

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

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

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