Chapter II. Complex Semisimple Lie Algebras

Abstract

The theme of this chapter is an investigation of complex semisimple Lie algebras by a two-step process, first by passing from such a Lie algebra to a reduced abstract root system via a choice of Cartan subalgebra and then by passing from the root system to an abstract Cartan matrix and an abstract Dynkin diagram via a choice of an ordering.

The chapter begins by making explicit a certain amount of this structure for four infinite classes of classical complex semisimple Lie algebras. Then for a general finite-dimensional complex Lie algebra, it is proved that Cartan subalgebras exist and are unique up to conjugacy.

When the given Lie algebra is semisimple, the Cartan subalgebra is abelian. The adjoint action of the Cartan subalgebra on the given semisimple Lie algebra leads to a root-space decomposition of the given Lie algebra, and the set of roots forms a reduced abstract root system.

If a suitable ordering is imposed on the underlying vector space of an abstract root system, one can define simple roots as those positive roots that are not sums of positive roots. The simple roots form a particularly nice basis of the underlying vector space, and a Cartan matrix and Dynkin diagram may be defined in terms of them. The definitions of abstract Cartan matrix and abstract Dynkin diagram are arranged so as to include the matrix and diagram obtained from a root system.

Use of the Weyl group shows that the Cartan matrix and Dynkin diagram obtained from a root system by imposing a ordering are in fact independent of the ordering. Moreover, nonisomorphic reduced abstract root systems have distinct Cartan matrices. It is possible to classify the abstract Cartan matrices and then to see by a case-by-case argument that every abstract Cartan matrix arises from a reduced abstract root system. Consequently the correspondence between reduced abstract root systems and abstract Cartan matrices is one-one onto, up to isomorphism.

The correspondence between complex semisimple Lie algebras and reduced abstract root systems lies deeper. Apart from isomorphism, the correspondence does not depend upon the choice of Cartan subalgebra, as a consequence of the conjugacy of Cartan subalgebras proved earlier in the chapter. To examine the correspondence more closely, one first finds generators and relations for any complex semisimple Lie algebra. The Isomorphism Theorem then explains how much freedom there is in lifting an isomorphism between root systems to an isomorphism between complex semisimple Lie algebras. Finally the Existence Theorem says that every reduced abstract root system arises from some complex semisimple Lie algebra. Consequently the correspondence between complex semisimple Lie algebras and reduced abstract root systems is one-one onto, up to isomorphism.