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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