Chapter 0

Review of Semisimple

Lie Algebras

The main technical prerequisite for this book is a good working knowledge

of the structure of semisimple Lie algebras over an algebraically closed field

of characteristic 0 such as C. (In fact, it is enough to work over a splitting

field of characteristic 0, as indicated below.) In this preliminary chapter

we summarize results with which the reader should be familiar, coupled

with some explicit references to the textbook literature. The basic structure

theory through the classification by Dynkin diagrams is treated in a very

large number of sources, of which we cite only a few here, e.g., Bourbaki

[45, 46], Carter [60], Humphreys [125, 129], Jacobson [143].

The subject can be approached from a number of angles, including the

traditional theory of Lie groups and the theory of linear algebraic groups;

but group theory generally remains in the background here. Since notation

varies considerably in the literature, the reader needs to be aware of our

conventions (which are often closest to those in [125]); these are intended to

steer something of a middle course among the available choices. Frequently

used notations are listed at the end of the book.

0.1. Cartan Decomposition

The basic object of study here is a semisimple Lie algebra g over a field of

characteristic 0, having a Cartan subalgebra f ) which is split: the eigenvalues

of ad/i are in the field, for all h £ f). Write £ := dimf). For convenience we

normally take C for the field, unless the contrary is stated.

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http://dx.doi.org/10.1090/gsm/094/01