Complex Semisimple Lie Groups

This chapter deals with the most explored section of the theory of Lie groups and Lie algebras. Its main result is the complete classification of connected complex semisimple Lie groups and their irreducible linear representations. This classification is based on the theory of root systems, which because of its numerous applications deserves a special treatment. The theory is axiomatically developed in §2. During the whole chapter (except 1.1°–1.3°) the ground field is ℂ. All the vector spaces and Lie algebras considered are finite-dimensional.