Lie groups 

In mathematics, a simple Lie group is a connected nonabelian 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.
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.
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 KillingCartan classification.
Unfortunately, there is no universally accepted definition of a simple Lie group. In particular, it is not always defined as a Lie group that is simple as an abstract group. Authors differ on whether a simple Lie group has to be connected, or on whether it is allowed to have a nontrivial center, or on whether is a simple Lie group.
The most common definition is that a Lie group is simple if it is connected, nonabelian, and every closed connected normal subgroup is either the identity or the whole group. In particular, simple groups are allowed to have a nontrivial center, but is not simple.
In this article the connected simple Lie groups with trivial center are listed. Once these are known, the ones with nontrivial center are easy to list as follows. Any simple Lie group with trivial center has a universal cover, whose center is the fundamental group of the simple Lie group. The corresponding simple Lie groups with nontrivial center can be obtained as quotients of this universal cover by a subgroup of the center.
An equivalent definition of a simple Lie group follows from the Lie correspondence: A connected Lie group is simple if its Lie algebra is simple. An important technical point is that a simple Lie group may contain discrete normal subgroups. For this reason, the definition of a simple Lie group is not equivalent to the definition of a Lie group that is 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.
As a counterexample, the general linear group is neither simple, nor semisimple. This is because multiples of the identity form a nontrivial normal subgroup, thus evading the definition. Equivalently, the corresponding Lie algebra has a degenerate Killing form, because multiples of the identity map to the zero element of the algebra. Thus, the corresponding Lie algebra is also neither simple nor semisimple. Another counterexample are the special orthogonal groups in even dimension. These have the matrix in the center, and this element is pathconnected to the identity element, and so these groups evade the definition. Both of these are reductive groups.
Main article: simple Lie algebra 
The Lie algebra of a simple Lie group is a simple Lie algebra. This is a onetoone correspondence between connected simple Lie groups with trivial center and simple Lie algebras of dimension greater than 1. (Authors differ on whether the onedimensional Lie algebra should be counted as simple.)
Over the complex numbers the semisimple Lie algebras are classified by their Dynkin diagrams, 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 forms 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.
Main article: 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 noncompact simple Lie group G, one compact and one noncompact. The noncompact 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 noncompact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry.
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 noncompact 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 III, B I and D I for p = 2, D III, and C I, and the two exceptional ones are types E III and E VII of complex dimensions 16 and 27.
stand for the real numbers, complex numbers, quaternions, and octonions.
In the symbols such as E_{6}^{−26} for the exceptional groups, the exponent −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).
Simple Lie groups are fully classified. The classification is usually stated in several steps, namely:
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
Main article: root system 
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, a 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. For those of type A and C we can find explicit matrix representations of the corresponding simply connected Lie group as matrix groups.
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_{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_{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. The symplectic groups have a doublecover by the metaplectic group.
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).
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_{'"`UNIQtemplatestyles0000002EQINU`"'7+1⁄2}.
See also: Abelian group 
Dimension  Outer automorphism group  Dimension of symmetric space  Symmetric space  Remarks  

(Abelian)  1  1  ^{†} 
See also: Compact group 
Dimension  Real rank  Fundamental group 
Outer automorphism group 
Other names  Remarks  

A_{n} (n ≥ 1) compact  n(n + 2)  0  Cyclic, order n + 1  1 if n = 1, 2 if n > 1.  projective special unitary group PSU(n + 1) 
A_{1} is the same as B_{1} and C_{1} 
B_{n} (n ≥ 2) compact  n(2n + 1)  0  2  1  special orthogonal group SO_{2n+1}(R) 
B_{1} is the same as A_{1} and C_{1}. B_{2} is the same as C_{2}. 
C_{n} (n ≥ 3) compact  n(2n + 1)  0  2  1  projective compact symplectic group PSp(n), PSp(2n), PUSp(n), PUSp(2n) 
Hermitian. Complex structures of H^{n}. Copies of complex projective space in quaternionic projective space. 
D_{n} (n ≥ 4) compact  n(2n − 1)  0  Order 4 (cyclic when n is odd).  2 if n > 4, S_{3} if n = 4  projective special orthogonal group PSO_{2n}(R) 
D_{3} is the same as A_{3}, D_{2} is the same as A_{1}^{2}, and D_{1} is abelian. 
E_{6}^{−78} compact  78  0  3  2  
E_{7}^{−133} compact  133  0  2  1  
E_{8}^{−248} compact  248  0  1  1  
F_{4}^{−52} compact  52  0  1  1  
G_{2}^{−14} compact  14  0  1  1  This is the automorphism group of the Cayley algebra. 
See also: Split Lie algebra 
Dimension  Real rank  Maximal compact subgroup 
Fundamental group 
Outer automorphism group 
Other names  Dimension of symmetric space 
Compact symmetric space 
NonCompact symmetric space 
Remarks  

A_{n} I (n ≥ 1) split  n(n + 2)  n  D_{n/2} or B_{(n−1)/2}  Infinite cyclic if n = 1 2 if n ≥ 2 
1 if n = 1 2 if n ≥ 2. 
projective special linear group PSL_{n+1}(R) 
n(n + 3)/2  Real structures on C^{n+1} or set of RP^{n} in CP^{n}. Hermitian if n = 1, in which case it is the 2sphere.  Euclidean structures on R^{n+1}. Hermitian if n = 1, when it is the upper half plane or unit complex disc.  
B_{n} I (n ≥ 2) split  n(2n + 1)  n  SO(n)SO(n+1)  Noncyclic, order 4  1  identity component of special orthogonal group SO(n,n+1) 
n(n + 1)  B_{1} is the same as A_{1}.  
C_{n} I (n ≥ 3) split  n(2n + 1)  n  A_{n−1}S^{1}  Infinite cyclic  1  projective symplectic group PSp_{2n}(R), PSp(2n,R), PSp(2n), PSp(n,R), PSp(n) 
n(n + 1)  Hermitian. Complex structures of H^{n}. Copies of complex projective space in quaternionic projective space.  Hermitian. Complex structures on R^{2n} compatible with a symplectic form. Set of complex hyperbolic spaces in quaternionic hyperbolic space. Siegel upper half space.  C_{2} is the same as B_{2}, and C_{1} is the same as B_{1} and A_{1}. 
D_{n} I (n ≥ 4) split  n(2n  1)  n  SO(n)SO(n)  Order 4 if n odd, 8 if n even  2 if n > 4, S_{3} if n = 4  identity component of projective special orthogonal group PSO(n,n) 
n^{2}  D_{3} is the same as A_{3}, D_{2} is the same as A_{1}^{2}, and D_{1} is abelian.  
E_{6}^{6} I split  78  6  C_{4}  Order 2  Order 2  E I  42  
E_{7}^{7} V split  133  7  A_{7}  Cyclic, order 4  Order 2  70  
E_{8}^{8} VIII split  248  8  D_{8}  2  1  E VIII  128  @ E8  
F_{4}^{4} I split  52  4  C_{3} × A_{1}  Order 2  1  F I  28  Quaternionic projective planes in Cayley projective plane.  Hyperbolic quaternionic projective planes in hyperbolic Cayley projective plane.  
G_{2}^{2} I split  14  2  A_{1} × A_{1}  Order 2  1  G I  8  Quaternionic subalgebras of the Cayley algebra. QuaternionKähler.  Nondivision quaternionic subalgebras of the nondivision Cayley algebra. QuaternionKähler. 
See also: Complex Lie group 
Real dimension  Real rank  Maximal compact subgroup 
Fundamental group 
Outer automorphism group 
Other names  Dimension of symmetric space 
Compact symmetric space 
NonCompact symmetric space  

A_{n} (n ≥ 1) complex  2n(n + 2)  n  A_{n}  Cyclic, order n + 1  2 if n = 1, 4 (noncyclic) if n ≥ 2.  projective complex special linear group PSL_{n+1}(C) 
n(n + 2)  Compact group A_{n}  Hermitian forms on C^{n+1}
with fixed volume. 
B_{n} (n ≥ 2) complex  2n(2n + 1)  n  B_{n}  2  Order 2 (complex conjugation)  complex special orthogonal group SO_{2n+1}(C) 
n(2n + 1)  Compact group B_{n}  
C_{n} (n ≥ 3) complex  2n(2n + 1)  n  C_{n}  2  Order 2 (complex conjugation)  projective complex symplectic group PSp_{2n}(C) 
n(2n + 1)  Compact group C_{n}  
D_{n} (n ≥ 4) complex  2n(2n − 1)  n  D_{n}  Order 4 (cyclic when n is odd)  Noncyclic of order 4 for n > 4, or the product of a group of order 2 and the symmetric group S_{3} when n = 4.  projective complex special orthogonal group PSO_{2n}(C) 
n(2n − 1)  Compact group D_{n}  
E_{6} complex  156  6  E_{6}  3  Order 4 (noncyclic)  78  Compact group E_{6}  
E_{7} complex  266  7  E_{7}  2  Order 2 (complex conjugation)  133  Compact group E_{7}  
E_{8} complex  496  8  E_{8}  1  Order 2 (complex conjugation)  248  Compact group E_{8}  
F_{4} complex  104  4  F_{4}  1  2  52  Compact group F_{4}  
G_{2} complex  28  2  G_{2}  1  Order 2 (complex conjugation)  14  Compact group G_{2} 
Dimension  Real rank  Maximal compact subgroup 
Fundamental group 
Outer automorphism group 
Other names  Dimension of symmetric space 
Compact symmetric space 
NonCompact symmetric space 
Remarks  

A_{2n−1} II (n ≥ 2) 
(2n − 1)(2n + 1)  n − 1  C_{n}  Order 2  SL_{n}(H), SU^{∗}(2n)  (n − 1)(2n + 1)  Quaternionic structures on C^{2n} compatible with the Hermitian structure  Copies of quaternionic hyperbolic space (of dimension n − 1) in complex hyperbolic space (of dimension 2n − 1).  
A_{n} III (n ≥ 1) p + q = n + 1 (1 ≤ p ≤ q) 
n(n + 2)  p  A_{p−1}A_{q−1}S^{1}  SU(p,q), A III  2pq  Hermitian. Grassmannian of p subspaces of C^{p+q}. If p or q is 2; quaternionKähler 
Hermitian. Grassmannian of maximal positive definite subspaces of C^{p,q}. If p or q is 2, quaternionKähler 
If p=q=1, split If p−q ≤ 1, quasisplit  
B_{n} I (n > 1) p+q = 2n+1 
n(2n + 1)  min(p,q)  SO(p)SO(q)  SO(p,q)  pq  Grassmannian of R^{p}s in R^{p+q}. If p or q is 1, Projective space If p or q is 2; Hermitian If p or q is 4, quaternionKähler 
Grassmannian of positive definite R^{p}s in R^{p,q}. If p or q is 1, Hyperbolic space If p or q is 2, Hermitian If p or q is 4, quaternionKähler 
If p−q ≤ 1, split.  
C_{n} II (n > 2) n = p+q (1 ≤ p ≤ q) 
n(2n + 1)  min(p,q)  C_{p}C_{q}  Order 2  1 if p ≠ q, 2 if p = q.  Sp_{2p,2q}(R)  4pq  Grassmannian of H^{p}s in H^{p+q}. If p or q is 1, quaternionic projective space in which case it is quaternionKähler. 
H^{p}s in H^{p,q}. If p or q is 1, quaternionic hyperbolic space in which case it is quaternionKähler. 

D_{n} I (n ≥ 4) p+q = 2n 
n(2n − 1)  min(p,q)  SO(p)SO(q)  If p and q ≥ 3, order 8.  SO(p,q)  pq  Grassmannian of R^{p}s in R^{p+q}. If p or q is 1, Projective space If p or q is 2 ; Hermitian If p or q is 4, quaternionKähler 
Grassmannian of positive definite R^{p}s in R^{p,q}. If p or q is 1, Hyperbolic Space If p or q is 2, Hermitian If p or q is 4, quaternionKähler 
If p = q, split If p−q ≤ 2, quasisplit  
D_{n} III (n ≥ 4) 
n(2n − 1)  ⌊n/2⌋  A_{n−1}R^{1}  Infinite cyclic  Order 2  SO^{*}(2n)  n(n − 1)  Hermitian. Complex structures on R^{2n} compatible with the Euclidean structure. 
Hermitian. Quaternionic quadratic forms on R^{2n}. 

E_{6}^{2} II (quasisplit) 
78  4  A_{5}A_{1}  Cyclic, order 6  Order 2  E II  40  QuaternionKähler.  QuaternionKähler.  Quasisplit but not split. 
E_{6}^{−14} III  78  2  D_{5}S^{1}  Infinite cyclic  Trivial  E III  32  Hermitian. Rosenfeld elliptic projective plane over the complexified Cayley numbers. 
Hermitian. Rosenfeld hyperbolic projective plane over the complexified Cayley numbers. 

E_{6}^{−26} IV  78  2  F_{4}  Trivial  Order 2  E IV  26  Set of Cayley projective planes in the projective plane over the complexified Cayley numbers.  Set of Cayley hyperbolic planes in the hyperbolic plane over the complexified Cayley numbers.  
E_{7}^{−5} VI  133  4  D_{6}A_{1}  Noncyclic, order 4  Trivial  E VI  64  QuaternionKähler.  QuaternionKähler.  
E_{7}^{−25} VII  133  3  E_{6}S^{1}  Infinite cyclic  Order 2  E VII  54  Hermitian.  Hermitian.  
E_{8}^{−24} IX  248  4  E_{7} × A_{1}  Order 2  1  E IX  112  QuaternionKähler.  QuaternionKähler.  
F_{4}^{−20} II  52  1  B_{4} (Spin_{9}(R))  Order 2  1  F II  16  Cayley projective plane. QuaternionKähler.  Hyperbolic Cayley projective plane. QuaternionKähler. 
The following table lists some Lie groups with simple Lie algebras of small dimension. The groups on a given line all have the same Lie algebra. In the dimension 1 case, the groups are abelian and not simple.
Dim  Groups  Symmetric space  Compact dual  Rank  Dim  

1  , S^{1} = U(1) = SO_{2}() = Spin(2)  Abelian  Real line  0  1  
3  S^{3} = Sp(1) = SU(2)=Spin(3), SO_{3}() = PSU(2)  Compact  
3  SL_{2}() = Sp_{2}(), SO_{2,1}()  Split, Hermitian, hyperbolic  Hyperbolic plane  Sphere S^{2}  1  2 
6  SL_{2}() = Sp_{2}(), SO_{3,1}(), SO_{3}()  Complex  Hyperbolic space  Sphere S^{3}  1  3 
8  SL_{3}()  Split  Euclidean structures on  Real structures on  2  5 
8  SU(3)  Compact  
8  SU(1,2)  Hermitian, quasisplit, quaternionic  Complex hyperbolic plane  Complex projective plane  1  4 
10  Sp(2) = Spin(5), SO_{5}()  Compact  
10  SO_{4,1}(), Sp_{2,2}()  Hyperbolic, quaternionic  Hyperbolic space  Sphere S^{4}  1  4 
10  SO_{3,2}(), Sp_{4}()  Split, Hermitian  Siegel upper half space  Complex structures on  2  6 
14  G_{2}  Compact  
14  G_{2}  Split, quaternionic  Nondivision quaternionic subalgebras of nondivision octonions  Quaternionic subalgebras of octonions  2  8 
15  SU(4) = Spin(6), SO_{6}()  Compact  
15  SL_{4}(), SO_{3,3}()  Split  ^{3} in ^{3,3}  Grassmannian G(3,3)  3  9 
15  SU(3,1)  Hermitian  Complex hyperbolic space  Complex projective space  1  6 
15  SU(2,2), SO_{4,2}()  Hermitian, quasisplit, quaternionic  ^{2} in ^{2,4}  Grassmannian G(2,4)  2  8 
15  SL_{2}(), SO_{5,1}()  Hyperbolic  Hyperbolic space  Sphere S^{5}  1  5 
16  SL_{3}()  Complex  SU(3)  2  8  
20  SO_{5}(), Sp_{4}()  Complex  Spin_{5}()  2  10  
21  SO_{7}()  Compact  
21  SO_{6,1}()  Hyperbolic  Hyperbolic space  Sphere S^{6}  
21  SO_{5,2}()  Hermitian  
21  SO_{4,3}()  Split, quaternionic  
21  Sp(3)  Compact  
21  Sp_{6}()  Split, hermitian  
21  Sp_{4,2}()  Quaternionic  
24  SU(5)  Compact  
24  SL_{5}()  Split  
24  SU_{4,1}  Hermitian  
24  SU_{3,2}  Hermitian, quaternionic  
28  SO_{8}()  Compact  
28  SO_{7,1}()  Hyperbolic  Hyperbolic space  Sphere S^{7}  
28  SO_{6,2}()  Hermitian  
28  SO_{5,3}()  Quasisplit  
28  SO_{4,4}()  Split, quaternionic  
28  SO^{∗}_{8}()  Hermitian  
28  G_{2}()  Complex  
30  SL_{4}()  Complex 
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.