Simple Lie group
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In mathematics, a simple Lie group is a connected non-abelian Lie group G such the quotient by the center is simple as an abstract group. An equivalent definition is that G is connected and has a non-abelian simple Lie algebra. (In particular a simple Lie group is not quite the same as a Lie group that is simple as an abstract group.) There are several inequivalent variations of this definition that are sometimes used.
A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself.
These groups, and groups closely related to them, include many of the so-called classical groups of geometry, which lie behind projective geometry and other geometries derived from it by the Erlangen programme of Felix Klein. They also include some exceptional groups, that were first discovered by those pursuing the classification of simple Lie groups. The exceptional groups account for many special examples and configurations in other branches of mathematics, as well contemporary theoretical physics. In particular the classification of finite simple groups depended on a thorough prior knowledge of the 'exceptional' possibilities.
The complete list of simple Lie groups is the basis for the theory of the semisimple Lie groups and reductive groups, and their representation theory. This has turned out not only to be a major extension of the theory of compact Lie groups (and their representation theory), but to be of basic significance in particle physics.
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[edit] Definition
Unfortunately there is no 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 0 dimensional Lie groups that are simple as abstract 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. Sometimes authors add the condition that the center is finite (or tivial); 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.
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.
[edit] 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.
[edit] 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.
[edit] 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.
[edit] 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:
[edit] Infinite series
[edit] A series
A1, A2, ...
Ar corresponds to the special unitary group, SU(r+1).
[edit] B series
B1, B2, ...
Br corresponds to the special orthogonal group, SO(2r+1).
[edit] C series
C1, C2, ...
Cr corresponds to the symplectic group, Sp(2r).
[edit] D series
D2, D3, ...
Dr 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.
[edit] Exceptional cases
For the exceptional cases see G2, F4, E6, E7, and E8.
See also E7½ (Lie algebra)
[edit] 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.
