Simple Lie group
In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces.
This article is about the Killing-Cartan classification. For a smaller list of groups that commonly occur in theoretical physics, see Table of Lie groups. For groups of dimension at most 3, see Bianchi classification.
Together with the commutative Lie group of the real numbers, , and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "blocks" that make up all (finite-dimensional) 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 so-called "special linear group" SL(n, ) of n by n matrices with determinant equal to 1 is simple for all odd n > 1, when it is isomorphic to the projective special linear group.
The first classification of simple Lie groups was by Wilhelm Killing, and this work was later perfected by Élie Cartan. The final classification is often referred to as Killing-Cartan classification.
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
Overview of the classification[edit]
Ar 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).
Br 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.
Cr 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. The symplectic groups have a double-cover by the metaplectic group.
Dr 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 D2 is two isolated nodes, the same as A1 ∪ A1, 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 D3 is the same as A3, corresponding to a covering map homomorphism from SU(4) to SO(6).
In addition to the four families Ai, Bi, Ci, and Di above, there are five so-called exceptional Dynkin diagrams G2, F4, E6, E7, and E8; 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 G2 is the automorphism group of the octonions, and the group associated to F4 is the automorphism group of a certain Albert algebra.
See also E7+1⁄2.
Simply laced groups[edit]
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.