List of simple 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 Killing-Cartan 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.
Simple Lie groups
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 non-trivial 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, non-abelian, and every closed connected normal subgroup is either the identity or the whole group. In particular, simple groups are allowed to have a non-trivial 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 non-trivial 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 non-trivial center can be obtained as quotients of this universal cover by a subgroup of the center.
Simple Lie algebras
The Lie algebra of a simple Lie group is a simple Lie algebra. This is a one-to-one correspondence between connected simple Lie groups with trivial center and simple Lie algebras of dimension greater than 1.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. 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
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.
Second, the product of symmetric spaces is symmetric, so we may as well just classify the irreducible simply connected ones.
The irreducible simply connected symmetric spaces are the real line, and exactly two symmetric spaces corresponding to each non-compact simple Lie group G,
one compact and one non-compact. The non-compact 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 non-compact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry.
Hermitian symmetric spaces
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 non-compact 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, D III, and C I, and the two exceptional ones are types E III and E VII of complex dimensions 16 and 27.
Notation
stand for the real numbers, complex numbers, quaternions, and octonions.In the symbols such as E6−26 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.
List
Abelian
| Dimension | Outer automorphism group | Dimension of symmetric space | Symmetric space | Remarks | |
| R | 1 | R∗ | 1 | R |
Compact
| Dimension | Real rank | Fundamental group | Outer automorphism group | Other names | Remarks | |
| An compact | n | 0 | Cyclic, order | 1 if, 2 if. | projective special unitary group | A1 is the same as B1 and C1 |
| Bn compact | n | 0 | 2 | 1 | special orthogonal group SO2n+1 | B1 is the same as A1 and C1. B2 is the same as C2. |
| Cn compact | n | 0 | 2 | 1 | projective compact symplectic group PSp, PSp, PUSp, PUSp | Hermitian. Complex structures of Hn. Copies of complex projective space in quaternionic projective space. |
| Dn compact | n | 0 | Order 4. | 2 if, S3 if | projective special orthogonal group PSO2n | D3 is the same as A3, D2 is the same as A12, and D1 is abelian. |
| E6−78 compact | 78 | 0 | 3 | 2 | ||
| E7−133 compact | 133 | 0 | 2 | 1 | ||
| E8−248 compact | 248 | 0 | 1 | 1 | ||
| F4−52 compact | 52 | 0 | 1 | 1 | ||
| G2−14 compact | 14 | 0 | 1 | 1 | This is the automorphism group of the Cayley algebra. |
Split
| Dimension | Real rank | Maximal compact subgroup | Fundamental group | Outer automorphism group | Other names | Dimension of symmetric space | Compact symmetric space | Non-Compact symmetric space | Remarks | |
| An I split | n | n | Dn/2 or B/2 | Infinite cyclic if n = 1 2 if n ≥ 2 | 1 if n = 1 2 if n ≥ 2. | projective special linear group PSLn+1 | n/2 | Real structures on Cn+1 or set of RPn in CPn. Hermitian if, in which case it is the 2-sphere. | Euclidean structures on Rn+1. Hermitian if, when it is the upper half plane or unit complex disc. | |
| Bn I split | n | n | SOSO | Non-cyclic, order 4 | 1 | identity component of special orthogonal group SO | n | B1 is the same as A1. | ||
| Cn I split | n | n | An−1S1 | Infinite cyclic | 1 | projective symplectic group PSp2n, PSp, PSp, PSp, PSp | n | Hermitian. Complex structures of Hn. Copies of complex projective space in quaternionic projective space. | Hermitian. Complex structures on R2n compatible with a symplectic form. Set of complex hyperbolic spaces in quaternionic hyperbolic space. Siegel upper half space. | C2 is the same as B2, and C1 is the same as B1 and A1. |
| Dn I split | n | n | SOSO | Order 4 if n odd, 8 if n even | 2 if, S3 if | identity component of projective special orthogonal group PSO | n2 | D3 is the same as A3, D2 is the same as A12, and D1 is abelian. | ||
| E66 I split | 78 | 6 | C4 | Order 2 | Order 2 | E I | 42 | |||
| E77 V split | 133 | 7 | A7 | Cyclic, order 4 | Order 2 | 70 | ||||
| E88 VIII split | 248 | 8 | D8 | 2 | 1 | E VIII | 128 | @ E8 | ||
| F44 I split | 52 | 4 | C3 × A1 | Order 2 | 1 | F I | 28 | Quaternionic projective planes in Cayley projective plane. | Hyperbolic quaternionic projective planes in hyperbolic Cayley projective plane. | |
| G22 I split | 14 | 2 | A1 × A1 | Order 2 | 1 | G I | 8 | Quaternionic subalgebras of the Cayley algebra. Quaternion-Kähler. | Non-division quaternionic subalgebras of the non-division Cayley algebra. Quaternion-Kähler. |
Complex
| Real dimension | Real rank | Maximal compact subgroup | Fundamental group | Outer automorphism group | Other names | Dimension of symmetric space | Compact symmetric space | Non-Compact symmetric space | |
| An complex | 2n | n | An | Cyclic, order | 2 if, 4 if. | projective complex special linear group PSLn+1 | n | Compact group An | Hermitian forms on Cn+1 with fixed volume. |
| Bn complex | 2n | n | Bn | 2 | Order 2 | complex special orthogonal group SO2n+1 | n | Compact group Bn | |
| Cn complex | 2n | n | Cn | 2 | Order 2 | projective complex symplectic group PSp2n | n | Compact group Cn | |
| Dn complex | 2n | n | Dn | Order 4 | Noncyclic of order 4 for, or the product of a group of order 2 and the symmetric group S3 when. | projective complex special orthogonal group PSO2n | n | Compact group Dn | |
| E6 complex | 156 | 6 | E6 | 3 | Order 4 | 78 | Compact group E6 | ||
| E7 complex | 266 | 7 | E7 | 2 | Order 2 | 133 | Compact group E7 | ||
| E8 complex | 496 | 8 | E8 | 1 | Order 2 | 248 | Compact group E8 | ||
| F4 complex | 104 | 4 | F4 | 1 | 2 | 52 | Compact group F4 | ||
| G2 complex | 28 | 2 | G2 | 1 | Order 2 | 14 | Compact group G2 |
Others
| Dimension | Real rank | Maximal compact subgroup | Fundamental group | Outer automorphism group | Other names | Dimension of symmetric space | Compact symmetric space | Non-Compact symmetric space | Remarks | |
| A2n−1 II | n − 1 | Cn | Order 2 | SLn, SU∗ | Quaternionic structures on C2n compatible with the Hermitian structure | Copies of quaternionic hyperbolic space in complex hyperbolic space. | ||||
| An III p + q = n + 1 | n | p | Ap−1Aq−1S1 | SU, A III | 2pq | Hermitian. Grassmannian of p subspaces of Cp+q. If p or q is 2; quaternion-Kähler | Hermitian. Grassmannian of maximal positive definite subspaces of Cp,q. If p or q is 2, quaternion-Kähler | If p=q=1, split If ≤ 1, quasi-split | ||
| Bn I p+q = 2n+1 | n | min | SOSO | SO | pq | Grassmannian of Rps in Rp+q. If p or q is 1, Projective space If p or q is 2; Hermitian If p or q is 4, quaternion-Kähler | Grassmannian of positive definite Rps in Rp,q. If p or q is 1, Hyperbolic space If p or q is 2, Hermitian If p or q is 4, quaternion-Kähler | If ≤ 1, split. | ||
| Cn II n = p+q | n | min | CpCq | Order 2 | 1 if p ≠ q, 2 if p = q. | Sp2p,2q | 4pq | Grassmannian of Hps in Hp+q. If p or q is 1, quaternionic projective space in which case it is quaternion-Kähler. | Hps in Hp,q. If p or q is 1, quaternionic hyperbolic space in which case it is quaternion-Kähler. | |
| Dn I p+q = 2n | n | min | SOSO | If p and q ≥ 3, order 8. | SO | pq | Grassmannian of Rps in Rp+q. If p or q is 1, Projective space If p or q is 2 ; Hermitian If p or q is 4, quaternion-Kähler | Grassmannian of positive definite Rps in Rp,q. If p or q is 1, Hyperbolic Space If p or q is 2, Hermitian If p or q is 4, quaternion-Kähler | If, split If ≤ 2, quasi-split | |
| Dn III | n | ⌊n/2⌋ | An−1R1 | Infinite cyclic | Order 2 | SO* | n | Hermitian. Complex structures on R2n compatible with the Euclidean structure. | Hermitian. Quaternionic quadratic forms on R2n. | |
| E62 II | 78 | 4 | A5A1 | Cyclic, order 6 | Order 2 | E II | 40 | Quaternion-Kähler. | Quaternion-Kähler. | Quasi-split but not split. |
| E6−14 III | 78 | 2 | D5S1 | 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. | |
| E6−26 IV | 78 | 2 | F4 | 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. | |
| E7−5 VI | 133 | 4 | D6A1 | Non-cyclic, order 4 | Trivial | E VI | 64 | Quaternion-Kähler. | Quaternion-Kähler. | |
| E7−25 VII | 133 | 3 | E6S1 | Infinite cyclic | Order 2 | E VII | 54 | Hermitian. | Hermitian. | |
| E8−24 IX | 248 | 4 | E7 × A1 | Order 2 | 1 | E IX | 112 | Quaternion-Kähler. | Quaternion-Kähler. | |
| F4−20 II | 52 | 1 | B4 | Order 2 | 1 | F II | 16 | Cayley projective plane. Quaternion-Kähler. | Hyperbolic Cayley projective plane. Quaternion-Kähler. |
Simple Lie groups of small dimension
The following table lists some Lie groups with simple Lie algebras of smalldimension. 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, S1=U=SO2=Spin | Abelian | Real line | 0 | 1 | |
| 3 | S3=Sp=SU=Spin, SO3=PSU | Compact | ||||
| 3 | SL2=Sp2, SO2,1 | Split, Hermitian, hyperbolic | Hyperbolic plane H2 | Sphere S2 | 1 | 2 |
| 6 | SL2=Sp2, SO3,1, SO3 | Complex | Hyperbolic space H3 | Sphere S3 | 1 | 3 |
| 8 | SL3 | Split | Euclidean structures on R3 | Real structures on C3 | 2 | 5 |
| 8 | SU | Compact | ||||
| 8 | SU | Hermitian, quasi-split, quaternionic | Complex hyperbolic plane | Complex projective plane | 1 | 4 |
| 10 | Sp=Spin, SO5 | Compact | ||||
| 10 | SO4,1, Sp2,2 | Hyperbolic, quaternionic | Hyperbolic space H4 | Sphere S4 | 1 | 4 |
| 10 | SO3,2,Sp4 | Split, Hermitian | Siegel upper half space | Complex structures on H2 | 2 | 6 |
| 14 | G2 | Compact | ||||
| 14 | G2 | Split, quaternionic | Non-division quaternionic subalgebras of non-division octonions | Quaternionic subalgebras of octonions | 2 | 8 |
| 15 | SU=Spin, SO6 | Compact | - | - | ||
| 15 | SL4, SO3,3 | Split | R3 in R3,3 | Grassmannian G | 3 | 9 |
| 15 | SU | Hermitian | Complex hyperbolic space | Complex projective space | 1 | 6 |
| 15 | SU, SO4,2 | Hermitian, quasi-split, quaternionic | R2 in R2,4 | Grassmannian G | 2 | 8 |
| 15 | SL2, SO5,1 | Hyperbolic | Hyperbolic space H5 | Sphere S5 | 1 | 5 |
| 16 | SL3 | Complex | SU | 2 | 8 | |
| 20 | SO5, Sp4 | Complex | Spin5 | 2 | 10 | |
| 21 | SO7 | Compact | - | - | ||
| 21 | SO6,1 | Hyperbolic | Hyperbolic space H6 | Sphere S6 | - | - |
| 21 | SO5,2 | Hermitian | - | - | ||
| 21 | SO4,3 | Split, quaternionic | - | - | ||
| 21 | Sp | Compact | - | - | ||
| 21 | Sp6 | Split, hermitian | - | - | ||
| 21 | Sp4,2 | Quaternionic | - | - | ||
| 24 | SU | Compact | - | - | ||
| 24 | SL5 | Split | - | - | ||
| 24 | SU4,1 | Hermitian | - | - | ||
| 24 | SU3,2 | Hermitian, quaternionic | - | - | ||
| 28 | SO8 | Compact | - | - | ||
| 28 | SO7,1 | Hyperbolic | Hyperbolic space H7 | Sphere S7 | - | - |
| 28 | SO6,2 | Hermitian | - | - | ||
| 28 | SO5,3 | Quasi-split | - | - | ||
| 28 | SO4,4 | Split, quaternionic | - | - | ||
| 28 | SO∗8 | Hermitian | - | - | ||
| 28 | G2 | Complex | - | - | ||
| 30 | SL4 | Complex | - | - |