List of simple Lie groups
Group theory → Lie groups Lie groups 


In mathematics, the simple Lie groups were first classified by Wilhelm Killing and later perfected by Élie Cartan; this classification is often referred to as KillingCartan classification.
The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces. See also the table of Lie groups for a smaller list of groups that commonly occur in theoretical physics, and the Bianchi classification for groups of dimension at most 3.
Contents
Simple Lie groups[edit]
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 R 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 R 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.
Simple Lie algebras[edit]
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.
Symmetric spaces[edit]
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.
Hermitian symmetric spaces[edit]
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.
Notation[edit]
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).
List[edit]
Abelian[edit]
Dimension  Outer automorphism group  Dimension of symmetric space  Symmetric space  Remarks  

R (Abelian)  1  R^{∗}  1  R  ^{†} 
Compact[edit]
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. 
Split[edit]
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. 
Complex[edit]
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} 
Others[edit]
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. 
Simple Lie groups of small dimension[edit]
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  R, S^{1}=U(1)=SO_{2}(R)=Spin(2)  Abelian  Real line  0  1  
3  S^{3}=Sp(1)=SU(2)=Spin(3), SO_{3}(R)=PSU(2)  Compact  
3  SL_{2}(R)=Sp_{2}(R), SO_{2,1}(R)  Split, Hermitian, hyperbolic  Hyperbolic plane H^{2}  Sphere S^{2}  1  2 
6  SL_{2}(C)=Sp_{2}(C), SO_{3,1}(R), SO_{3}(C)  Complex  Hyperbolic space H^{3}  Sphere S^{3}  1  3 
8  SL_{3}(R)  Split  Euclidean structures on R^{3}  Real structures on C^{3}  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}(R)  Compact  
10  SO_{4,1}(R), Sp_{2,2}(R)  Hyperbolic, quaternionic  Hyperbolic space H^{4}  Sphere S^{4}  1  4 
10  SO_{3,2}(R),Sp_{4}(R)  Split, Hermitian  Siegel upper half space  Complex structures on H^{2}  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}(R)  Compact  
15  SL_{4}(R), SO_{3,3}(R)  Split  R^{3} in R^{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}(R)  Hermitian, quasisplit, quaternionic  R^{2} in R^{2,4}  Grassmannian G(2,4)  2  8 
15  SL_{2}(H), SO_{5,1}(R)  Hyperbolic  Hyperbolic space H^{5}  Sphere S^{5}  1  5 
16  SL_{3}(C)  Complex  SU(3)  2  8  
20  SO_{5}(C), Sp_{4}(C)  Complex  Spin_{5}(R)  2  10  
21  SO_{7}(R)  Compact  
21  SO_{6,1}(R)  Hyperbolic  Hyperbolic space H^{6}  Sphere S^{6}  
21  SO_{5,2}(R)  Hermitian  
21  SO_{4,3}(R)  Split, quaternionic  
21  Sp(3)  Compact  
21  Sp_{6}(R)  Split, hermitian  
21  Sp_{4,2}(R)  Quaternionic  
24  SU(5)  Compact  
24  SL_{5}(R)  Split  
24  SU_{4,1}  Hermitian  
24  SU_{3,2}  Hermitian, quaternionic  
28  SO_{8}(R)  Compact  
28  SO_{7,1}(R)  Hyperbolic  Hyperbolic space H^{7}  Sphere S^{7}  
28  SO_{6,2}(R)  Hermitian  
28  SO_{5,3}(R)  Quasisplit  
28  SO_{4,4}(R)  Split, quaternionic  
28  SO^{∗}_{8}(R)  Hermitian  
28  G_{2}(C)  Complex  
30  SL_{4}(C)  Complex 
Notes[edit]
 ^† The group R is not simple as an abstract group, and according to most (but not all) definitions this is not a simple Lie group. Most authors do not count its Lie algebra as a simple Lie algebra, it is listed here so that the list of irreducible simply connected symmetric spaces is complete. Note that R is the only such noncompact symmetric space without a compact dual (although it has a compact quotient S^{1}).
Further reading[edit]
 Besse, Einstein manifolds ISBN 0387152792
 Helgason, Differential geometry, Lie groups, and symmetric spaces. ISBN 0821828487
 Fuchs and Schweigert, Symmetries, Lie algebras, and representations: a graduate course for physicists. Cambridge University Press, 2003. ISBN 0521541190