Simple Lie group
In mathematics, a simple Lie group is a connected nonabelian Lie group G which does not have nontrivial connected normal subgroups.
Group theory → Lie groups Lie groups 


Together with the commutative Lie group of the real numbers, , and that of the unitmagnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "blocks" that make up all (finitedimensional) connected Lie groups via the operation of group extension. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the socalled "special linear group" SL(n) of n by n matrices with determinant equal to 1 is simple for all n > 1.
An equivalent definition of a simple Lie group follows from the Lie correspondence: a connected Lie group is simple if its Lie algebra is a simple. An important technical point is that a simple Lie group may contain discrete normal subgroups, hence being a simple Lie group is different from being simple as an abstract group.
Simple Lie groups include many classical Lie groups, which provide a grouptheoretic underpinning for spherical geometry, projective geometry and related geometries in the sense of Felix Klein's Erlangen program. It emerged in the course of classification of simple Lie groups that there exist also several exceptional possibilities not corresponding to any familiar geometry. These exceptional groups account for many special examples and configurations in other branches of mathematics, as well as contemporary theoretical physics.
Classification of simple Lie groups
Full classification
Simple Lie groups are fully classified. The classification is usually stated in several steps, namely:
 Classification of simple complex Lie algebras The classification of simple Lie algebras over the complex numbers by Dynkin diagrams.
 Classification of simple real Lie algebras Each simple complex Lie algebra has several real forms, classified by additional decorations of its Dynkin diagram called Satake diagrams, after Ichirô Satake.
 Classification of centerless simple Lie groups For every (real or complex) simple Lie algebra , there is a unique "centerless" simple Lie group whose Lie algebra is and which has trivial center.
 Classification of simple Lie groups
One can show that the fundamental group of any Lie group is a discrete commutative group. Given a (nontrivial) subgroup of the fundamental group of some Lie group , one can use the theory of covering spaces to construct a new group with in its center. Now any (real or complex) Lie group can be obtained by applying this construction to centerless Lie groups. Note that real Lie groups obtained this way might not be real forms of any complex group. A very important example of such a real group is the metaplectic group, which appears in infinitedimensional representation theory and physics. When one takes for the full fundamental group, the resulting Lie group is the universal cover of the centerless Lie group , and is simply connected. In particular, every (real or complex) Lie algebra also corresponds to a unique connected and simply connected Lie group with that Lie algebra, called the "simply connected Lie group" associated to
Compact Lie groups
Every simple complex Lie algebra has a unique real form whose corresponding centerless Lie group is compact. It turns out that the simply connected Lie group in these cases is also compact. Compact Lie groups have a particularly tractable representation theory because of the Peter–Weyl theorem. Just like simple complex Lie algebras, centerless compact Lie groups are classified by Dynkin diagrams (first classified by Wilhelm Killing and Élie Cartan).
For the infinite (A, B, C, D) series of Dynkin diagrams, the simply connected compact Lie group associated to each Dynkin diagram can be explicitly described as a matrix group, with the corresponding centerless compact Lie group described as the quotient by a subgroup of scalar matrices.
A series
A_{1}, A_{2}, ...
A_{r} has as its associated simply connected compact group the special unitary group, SU(r + 1) and as its associated centerless compact group the projective unitary group PU(r + 1).
B series
B_{2}, B_{3}, ...
B_{r} has as its associated centerless compact groups the odd special orthogonal groups, SO(2r + 1). This group is not simply connected however: its universal (double) cover is the Spin group.
C series
C_{3}, C_{4}, ...
C_{r} has as its associated simply connected group the group of unitary symplectic matrices, Sp(r) and as its associated centerless group the Lie group PSp(r) = Sp(r)/{I, −I} of projective unitary symplectic matrices.
D series
D_{4}, D_{5}, ...
D_{r} has as its associated compact group the even special orthogonal groups, SO(2r) and as its associated centerless compact group the projective special orthogonal group PSO(2r) = SO(2r)/{I, −I}. As with the B series, SO(2r) is not simply connected; its universal cover is again the spin group, but the latter again has a center (cf. its article).
The diagram D_{2} is two isolated nodes, the same as A_{1} ∪ A_{1}, and this coincidence corresponds to the covering map homomorphism from SU(2) × SU(2) to SO(4) given by quaternion multiplication; see quaternions and spatial rotation. Thus SO(4) is not a simple group. Also, the diagram D_{3} is the same as A_{3}, corresponding to a covering map homomorphism from SU(4) to SO(6).
Exceptional cases
In addition to the four families A_{i}, B_{i}, C_{i}, and D_{i} above, there are five socalled exceptional Dynkin diagrams G_{2}, F_{4}, E_{6}, E_{7}, and E_{8}; these exceptional Dynkin diagrams also have associated simply connected and centerless compact groups. However, the groups associated to the exceptional families are more difficult to describe than those associated to the infinite families, largely because their descriptions make use of exceptional objects. For example, the group associated to G_{2} is the automorphism group of the octonions, and the group associated to F_{4} is the automorphism group of a certain Albert algebra.
See also E_{7½}.
Simply laced groups
A simply laced group is a Lie group whose Dynkin diagram only contain simple links, and therefore all the nonzero roots of the corresponding Lie algebra have the same length. The A, D and E series groups are all simply laced, but no group of type B, C, F, or G is simply laced.
References
 Jacobson, Nathan (1971). Exceptional Lie Algebras (1 ed.). CRC Press. ISBN 0824713265.
 Fulton, WIlliam and Harris, Joe (2004). Representation Theory: A First Course. Springer.doi:10.1007/9781461209799