Proceedings of Symposia in Pure Mathematics
Volume 61 (1997), pp. 1-27
Structure Theory of Semisimple Lie Groups
A. W. Knapp
This article provides a review of the elementary theory of semisimple Lie al-
gebras and Lie groups. It is essentially a summary of much of [K3]. The four
sections treat complex semisimple Lie algebras, finite-dimensional representations
of complex semisimple Lie algebras, compact Lie groups and real forms of complex
Lie algebras, and structure theory of noncompact semisimple groups.
1. Complex Semisimple Lie Algebras
This section deals with the structure theory of complex semisimple Lie algebras.
Some references for this material are [He], [Hu], [J], [Kl], [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 r&dg be the sum of all the
solvable ideals in g. The sum of two solvable ideals is a solvable ideal [K3, §1.2],
and the finite-dimensionality of g makes radg a solvable ideal. We say that g is
semisimple if radg = 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 ad Y). It is invariant in the sense that B([X, Y],Z) = B(X, [Y, Z\) for all
X,Y,Z in g.
Theorem 1.1 (Cartan's criterion for semisimplicity). The Lie algebra g is
semisimple if and only if B is nondegenerate.
REFERENCE. [K3, Theorem 1.42].
The Lie algebra g is said to be simple if g is nonabelian and g has no proper
nonzero ideals. In this case, [g,g] = g. Semisimple Lie algebras and simple Lie
algebras are related as in the following theorem.
Theorem 1.2. The Lie algebra g is semisimple if and only if g is the direct
sum of simple ideals. In this case there are no other simple ideals, the direct sum
decomposition is unique up to the order of the summands, and every ideal is the
sum of some subset of the simple ideals. Also in this case, [9,9] = g.
1991 Mathematics Subject Classification. Primary 17B20, 20G05, 22E15.
©1997 A. W. Knapp