Exceptional simple Lie group

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. A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself. A direct sum of simple Lie algebras is called a semisimple Lie algebra.

An equivalent definition of a simple Lie group follows from the Lie correspondence: a connected Lie group is simple if its Lie algebra is non-abelian and 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 group-theoretic underpinning for spherical geometry, projective geometry and related geometries in the sense of Felix Klein's Erlangen programme. 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.

While the notion of a simple Lie group is satisfying from the axiomatic perspective, in applications of Lie theory, such as the theory of Riemannian symmetric spaces, somewhat more general notions of semisimple and reductive Lie groups proved to be even more useful. In particular, every connected compact Lie group is reductive, and the study of representations of general reductive groups is a major branch of representation theory.

Comments on the definition

Unfortunately there is no single standard definition of a simple Lie group. The definition given above is sometimes varied in the following ways:

Connectedness: Usually simple Lie groups are connected by definition. This excludes discrete simple groups (these are zero-dimensional Lie groups that are simple as abstract groups) as well as disconnected orthogonal groups.
Center: Usually simple Lie groups are allowed to have a discrete center; for example, SL₂(R) has a center of order 2, but is still counted as a simple Lie group. If the center is non-trivial (and not the whole group) then the simple Lie group is not simple as an abstract group. Some authors require that the center of a simple Lie group be finite (or trivial); the universal cover of SL₂(R) is an example of a simple Lie group with infinite center.
R: Usually the group R of real numbers under addition (and its quotient R/Z) are not counted as simple Lie groups, even though they are connected and have a Lie algebra with no proper non-zero ideals. Occasionally authors define simple Lie groups in such a way that R is simple, though this sometimes seems to be an accident caused by overlooking this case.
Matrix groups: Some authors restrict themselves to Lie groups that can be represented as groups of finite matrices. The metaplectic group is an example of a simple Lie group that cannot be represented in this way.
Complex Lie algebras: The definition of a simple Lie algebra is not stable under the extension of scalars. The complexification of a complex simple Lie algebra, such as sl_n(C) is semisimple, but not simple.

The most common definition is the one above: simple Lie groups have to be connected, they are allowed to have non-trivial centers (possibly infinite), they need not be representable by finite matrices, and they must be non-abelian.

Method of classification

Such groups are classified using the prior classification of the complex simple Lie algebras: for which see the page on root systems. It is shown that a simple Lie group has a simple Lie algebra that will occur on the list given there, once it is complexified (that is, made into a complex vector space rather than a real one). This reduces the classification to two further matters.

Real forms

The groups SO(p,q,R) and SO(p+q,R), for example, give rise to different real Lie algebras, but having the same Dynkin diagram. In general there may be different real forms of the same complex Lie algebra.

Relationship of simple Lie algebras to groups

Secondly the Lie algebra only determines uniquely the simply connected (universal) cover G* of the component containing the identity of a Lie group G. It may well happen that G* isn't actually a simple group, for example having a non-trivial center. We have therefore to worry about the global topology, by computing the fundamental group of G (an abelian group: a Lie group is an H-space). This was done by Élie Cartan.

For an example, take the special orthogonal groups in even dimension. With the non-identity matrix −I in the center, these aren't actually simple groups; and having a two-fold spin cover, they aren't simply-connected either. They lie 'between' G* and G, in the notation above.

Classification by Dynkin diagram

See main article root system

According to Dynkin's classification, we have as possibilities these only, where n is the number of nodes:

Infinite series

A series

A₁, A₂, ...

A_r corresponds to the special unitary group, SU(r+1).

B series

B₁, B₂, ...

B_r corresponds to the special orthogonal group, SO(2r+1).

C series

C₁, C₂, ...

C_r corresponds to the symplectic group, Sp(2r).

D series

D₂, D₃, ...

D_r corresponds to the special orthogonal group, SO(2r). Note that SO(4) is not a simple group, though. The Dynkin diagram has two nodes that are not connected. There is a surjective homomorphism from SO(3)* × SO(3)* to SO(4) given by quaternion multiplication; see quaternions and spatial rotation. Therefore the simple groups here start with D₃, which as a diagram straightens out to A₃. With D₄ there is an 'exotic' symmetry of the diagram, corresponding to so-called triality.

Exceptional cases

For the so-called exceptional cases see G₂, F₄, E₆, E₇, and E₈. These cases are deemed 'exceptional' because they do not fall into infinite series of groups of increasing dimension. From the point of view of each group taken separately, there is nothing so unusual about them. These exceptional groups were discovered around 1890 in the classification of the simple Lie algebras, over the complex numbers (Wilhelm Killing, re-done by Élie Cartan). For some time it was a research issue to find concrete ways in which they arise, for example as a symmetry group of a differential system.

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.