Structure Theory of Semisimple Lie Groups

This section deals with the structure theory of complex semisimple Lie algebras. Some references for this material are [He], [Hu], [J], [K1], [K3], and [V]. Let g be a finite-dimensional Lie algebra. For the moment we shall allow the underlying field to be R or C, but shortly we shall restrict to Lie algebras over C. Semisimple Lie algebras are defined as follows. Let rad g be the sum of all the solvable ideals in g. The sum of two solvable ideals is a solvable ideal [K3, §I.2], and the finite-dimensionality of g makes rad g a solvable ideal. We say that g is semisimple if rad g = 0. Within g, let adX be the linear transformation given by (adX)Z = [X,Z]. The Killing form is the symmetric bilinear form on g defined by B(X,Y ) = Tr(adX adY ). It is invariant in the sense that B([X,Y ], Z) = B(X, [Y,Z]) for all X,Y, Z in g.