##
**AlgĂ¨bres de Lie semi-simples complexes.**
*(French)*
Zbl 0144.02105

New York-Amsterdam: W. A. Benjamin, Inc. viii, 130 p. (not consecutively paged) (1966).

In this lecture note, the author exposes the basic theorems on complex semisimple Lie algebras in a concise and lucid style. In the first two chapters, the author gives the well known general properties on nilpotent, solvable and semi-simple Lie algebras without proofs. [As for these results, for example, one can refer to his previous lecture note [Lie algebras and Lie groups. 1964 lectures given at Harvard University. New York etc.: W. A. Benjamin, Inc. (1965; Zbl 0132.27803).]

In the following chapters, he deals with the theory of semisimple Lie algebras. Cartan subalgebras are defined as a nilpotent subalgebra which is equal to its own normalizer and its fundamental properties are proved. Root systems are defined by the method of Bourbaki. It contains the non-reduced root systems which appear when one supposes the base field is not algebraically closed. Then, he gives the existence and uniqueness theorem of semisimple Lie algebras corresponding to a given reduced root system.

As for the representation theory, giving detailed discussion on the algebra \(\mathfrak{sl}_2\), he shows the correspondence between irreducible modules of finite dimensions and the dominant weights and Weyl’s formula for the dimension of a representation space.

In the last chapter, he gives, without proofs, the structure of the complex and compact Lie groups related to the structure of Lie algebras. This gives a simple and good introduction to the topology of Lie groups and to the theory of algebraic groups.

In the following chapters, he deals with the theory of semisimple Lie algebras. Cartan subalgebras are defined as a nilpotent subalgebra which is equal to its own normalizer and its fundamental properties are proved. Root systems are defined by the method of Bourbaki. It contains the non-reduced root systems which appear when one supposes the base field is not algebraically closed. Then, he gives the existence and uniqueness theorem of semisimple Lie algebras corresponding to a given reduced root system.

As for the representation theory, giving detailed discussion on the algebra \(\mathfrak{sl}_2\), he shows the correspondence between irreducible modules of finite dimensions and the dominant weights and Weyl’s formula for the dimension of a representation space.

In the last chapter, he gives, without proofs, the structure of the complex and compact Lie groups related to the structure of Lie algebras. This gives a simple and good introduction to the topology of Lie groups and to the theory of algebraic groups.

Reviewer: Eiichi Abe (Ibaraki)

### MSC:

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

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

17B22 | Root systems |

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

17B30 | Solvable, nilpotent (super)algebras |