Editing Simple Lie group (section)

==Related ideas==
===Semisimple Lie groups===
A '''semisimple''' Lie group is a connected Lie group so that its only [[closed subgroup|closed]] [[Connected space|connected]] [[Abelian group|abelian]] [[normal subgroup|normal]] subgroup is the trivial subgroup. Every simple Lie group is semisimple. More generally, any product of simple Lie groups is semisimple, and any quotient of a semisimple Lie group by a closed subgroup is semisimple. Every semisimple Lie group can be formed by taking a product of simple Lie groups and quotienting by a subgroup of its center. In other words, every semisimple Lie group is a [[central product]] of simple Lie groups. The semisimple Lie groups are exactly the Lie groups whose Lie algebras are [[semisimple Lie algebra]]s.

===Simple Lie algebras===
{{main|simple Lie algebra}}
The [[Lie group–Lie algebra correspondence|Lie algebra of a simple Lie group]] is a simple Lie algebra. This is a one-to-one correspondence between connected simple Lie groups with [[Triviality (mathematics)|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 semisimple Lie algebras are classified by their [[Dynkin diagram]]s, of types "ABCDEFG". 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 form]]s 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.

===Symmetric spaces===
{{Main|Symmetric space#Classification of Riemannian symmetric spaces}}
Symmetric spaces are classified as follows.

First, the universal cover of a symmetric space is still symmetric, so we can reduce to the case of simply connected symmetric spaces. (For example, the universal cover of a real projective plane is a sphere.)

Second, the product of symmetric spaces is symmetric, so we may as well just classify the irreducible simply connected ones (where irreducible means they cannot be written as a product of smaller symmetric spaces).

The irreducible simply connected symmetric spaces are the real line, and exactly two symmetric spaces corresponding to  each ''non-compact'' simple Lie group ''G'',
one compact and one non-compact. The non-compact one is a cover of the quotient of ''G'' by a maximal compact subgroup ''H'', and the compact one is a cover of the quotient of
the compact form of ''G'' by the same subgroup ''H''. This duality between compact and non-compact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry.

===Hermitian symmetric spaces===

A symmetric space with a compatible complex structure is called Hermitian.  The compact simply connected irreducible Hermitian symmetric spaces fall into 4 infinite families with 2 exceptional ones left over, and each has a non-compact dual. In addition the complex plane is also a Hermitian symmetric space; this gives the complete list of irreducible Hermitian symmetric spaces.

The four families are the types A&nbsp;III, B&nbsp;I and D&nbsp;I for {{nowrap|1=''p'' = 2}}, D&nbsp;III, and C&nbsp;I, and the two exceptional ones are types E&nbsp;III and E&nbsp;VII of complex dimensions 16 and 27.

===Notation===
<math> \mathbb {R, C, H, O} </math>&nbsp; stand for the real numbers, complex numbers, [[quaternions]], and [[octonion]]s.

In the symbols such as ''E''<sub>6</sub><sup>&minus;26</sup> for the exceptional groups, the exponent &minus;26 is the signature of an invariant symmetric bilinear form that is negative definite on the maximal compact subgroup. It is equal to the dimension of the group minus twice the dimension of a maximal compact subgroup.

The fundamental group listed in the table below is the fundamental group of the simple group with trivial center. 
Other simple groups with the same Lie algebra correspond to subgroups of this fundamental group (modulo the action of the outer automorphism group).