List of Simple Lie Groups - Simple Lie Algebras

Simple Lie Algebras

The Lie algebra of a simple Lie group is a simple Lie algebra, and this gives a one to one correspondence between connected simple Lie groups with trivial center and simple Lie algebras of dimension greater than 1. (Authors differ on whether the one dimensional Lie algebra should be counted as simple.)

Over the complex numbers the simple Lie algebras are given by the usual "ABCDEFG" classification. If L is a real simple Lie algebra, its complexification is a simple complex Lie algebra, unless L is already the complexification of a Lie algebra, in which case the complexification of L is a product of two copies of L. This reduces the problem of classifying the real simple Lie algebras to that of finding all the real forms of each complex simple Lie algebra (i.e., real Lie algebras whose complexification is the given complex Lie algebra). There are always at least 2 such forms: a split form and a compact form, and there are usually a few others. The different real forms correspond to the classes of automorphisms of order at most 2 of the complex Lie algebra.