an:03231961
Zbl 0144.02105
Serre, Jean-Pierre
Algèbres de Lie semi-simples complexes
FR
New York-Amsterdam: W. A. Benjamin, Inc. viii, 130 p. (not consecutively paged) (1966).
1966
b
17-02 17B20 17B22 17B10 17B30
complex semisimple Lie algebras; Cartan subalgebras; root systems; representation theory; dominant weights; Weyl's formula; dimension of representation space
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.
Eiichi Abe (Ibaraki)
0132.27803