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.
Read more about this topic: List Of Simple Lie Groups
Famous quotes containing the words simple and/or lie:
“But the whim we have of happiness is somewhat thus. By certain valuations, and averages, of our own striking, we come upon some sort of average terrestrial lot; this we fancy belongs to us by nature, and of indefeasible rights. It is simple payment of our wages, of our deserts; requires neither thanks nor complaint.... Foolish soul! What act of legislature was there that thou shouldst be happy? A little while ago thou hadst no right to be at all.”
—Thomas Carlyle (1795–1881)
“‘Tis to yourself I speak; you cannot know
Him whom I call in speaking such a one,
For you beneath the earth lie buried low,
Which he alone as living walks upon:”
—Jones Very (1831–1880)