Definitions

Exceptional Lie groups

List of simple Lie groups

In mathematics, the simple Lie groups were classified by Élie Cartan. 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 generally accepted definition of a simple Lie group, and in particular it is not necessarily 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 implies that simple Lie groups must be connected, and non-abelian, but are allowed to have a non-trivial center.

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, and this gives a one to one correspondence between connected simple Lie groups with trivial center and simple Lie algebras of dimension greater than 1. (Authors differ on whether the one dimensional Lie algebra should be counted as simple.)

Over the complex numbers the simple Lie algebras are given by the usual "ABCDEFG" classification. 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. (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 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 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

R, C, H, and O 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 (modulo the action of the outer automorphism group).

The list of simple Lie groups

R (Abelian)

Dimension: 1

Outer automorphism group: R*

Dimension of symmetric space: 1

Symmetric space: R

Remarks: This group 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 non-compact symmetric space without a compact dual (although of course it has a compact quotient S1).

An (n ≥ 1) compact

Dimension: n(n + 2)

Real rank: 0

Fundamental group: Cyclic, order n + 1

Outer automorphism group: 1 if n = 1, 2 if n > 1.

Other names: PSU(n + 1), projective special unitary group.

Remarks: A1 is the same as the compact forms of B1 and C1

An I (n ≥ 1) (split)

Dimension: n(n + 2)

Real rank: n

Maximal compact subgroup: Dn/2 or B(n−1)/2

Fundamental group: 2 if n ≥ 2, infinite cyclic if n = 1.

Outer automorphism group: 1 if n = 1, 2 if n ≥ 2.

Other names: PSLn+1(R), projective special linear group.

Dimension of symmetric space: n(n + 3)/2

Compact symmetric space: Real structures on Cn+1 or set of RPn in CPn. Hermitian if n = 1, in which case it is the sphere.

Non-compact symmetric space: Euclidean structures on Rn+1. Hermitian if n = 1, when it is the upper half plane or unit complex disc.

Remarks:

A2n−1 II (n ≥ 2)

Dimension: (2n − 1)(2n + 1)

Real rank: n − 1

Maximal compact subgroup: Cn

Fundamental group:

Outer automorphism group:

Other names: SLn(H), SU*(2n)

Dimension of symmetric space: (n − 1)(2n + 1)

Compact symmetric space: Quaternionic structures on C2n compatible with the Hermitian structure.

Non-compact symmetric space: Copies of quaternionic hyperbolic space (of dimension n − 1) in complex hyperbolic space (of dimension 2n − 1).

Remarks:

An III (n ≥ 1, p + q = n + 1, 1 ≤ pq)

Dimension: n(n + 2)

Real rank: p

Maximal compact subgroup: Ap−1Aq−1S1

Fundamental group:

Outer automorphism group:

Other names: SU(p,q), A III

Dimension of symmetric space: 2pq

Compact symmetric space: Hermitian. Quaternion-Kähler if p or q is 2. Grassmannian of p subspaces of Cp+q.

Non-compact symmetric space: Hermitian. Quaternion-Kähler if p or q is 2. Grassmannian of maximal positive definite subspaces of Cp,q.

Remarks:

An (n ≥ 1) complex

Dimension: 2n(n + 2)

Real rank: n

Maximal compact subgroup: An

Fundamental group: Cyclic, order n + 1

Outer automorphism group: 2 if n = 1, 4 (noncyclic) if n ≥ 2.

Other names: PSLn+1(C), complex projective special linear group.

Dimension of symmetric space: n(n + 2)

Compact symmetric space: Compact group An

Non-compact symmetric space: Hermitian forms on Cn+1 with fixed volume.

Remarks:

Bn (n > 1) compact

Dimension: n(2n + 1)

Real rank: 0

Fundamental group: 2

Outer automorphism group: 1

Other names: SO2n+1(R), special orthogonal group.

Remarks: B1 is the same as A1 and C1. B2 is the same as C2.

Bn I (n > 1)

Dimension: n(2n + 1)

Real rank: min(p,q)

Maximal compact subgroup:

Fundamental group:

Outer automorphism group:

Other names: SO(p,q)

Dimension of symmetric space: pq

Compact symmetric space: Grassmannian of Rps in Rp+q. This is projective space if p or q is 1. Quaternion-Kähler if p or q is 4. Hermitian if p or q is 2.

Non-compact symmetric space: Grassmannian of positive definite Rps in Rp,q. This is hyperbolic space if p or q is 1. Quaternion-Kähler if p or q is 4. Hermitian if p or q is 2.

Remarks:

Bn (n > 1) complex

Dimension: 2n(2n + 1)

Real rank: n

Maximal compact subgroup: Bn

Fundamental group: 2

Outer automorphism group: order 2 (complex conjugation)

Other names:

Dimension of symmetric space: n(2n + 1)

Compact symmetric space: Compact group Bn

Non-compact symmetric space:

Remarks:

Cn (n ≥ 3) compact

Dimension: n(2n + 1)

Real rank: 0

Fundamental group: 2

Outer automorphism group: 1

Other names: Sp(n), Sp(2n), USp(n), USp(2n)

Remarks: C1 is the same as B1 and A1. C2 is the same as B2.

Cn I (n ≥ 3) (split)

Dimension: n(2n + 1)

Real rank: n

Maximal compact subgroup: An−1S1

Fundamental group: infinite cyclic

Outer automorphism group: 1

Other names: Symplectic group, Sp2n(R), Sp(2n,R),Sp(2n), Sp(n,R), Sp(n)

Dimension of symmetric space: n(n + 1)

Compact symmetric space: Hermitian. Complex structures of Hn. Copies of complex projective space in quaternionic projective space.

Non-compact symmetric space: Hermitian. Lagrangian subspaces of R2n. Set of complex hyperbolic spaces in quaternionic hyperbolic space. Siegel upper half plane.

Remarks: C2 is the same as B2, and C1 is the same as B1 and A1.

Cn II (n > 2, n = p + q, 1 ≤ pq)

Dimension: n(2n + 1)

Real rank: min(p,q)

Maximal compact subgroup: CpCq

Fundamental group: Order 2

Outer automorphism group: Trivial unless p=q, in which case it has order 2.

Other names: Sp2p,2q(R)

Dimension of symmetric space: 4pq

Compact symmetric space: Grassmannian of Hps in Hp+q. Quaternionic projective space if p or q is 1, in which case it is Quaternion-Kähler.

Non-compact symmetric space: Grassmannian of positive definite Hps in Hp,q. Quaternionic hyperbolic space if p or q is 1, in which case it is Quaternion-Kähler.

Remarks:

Cn (n > 2) complex

Dimension: 2n(2n + 1)

Real rank: n

Maximal compact subgroup: Cn

Fundamental group: 2

Outer automorphism group: Order 2 (complex conjugation)

Other names: Complex symplectic group, Sp2n(C)

Dimension of symmetric space: n(2n + 1)

Compact symmetric space: Compact group Cn

Non-compact symmetric space:

Remarks:

Dn (n ≥ 4) compact

Dimension: n(2n − 1)

Real rank: 0

Fundamental group: Order 4, (cyclic when n is odd).

Outer automorphism group: 2 if n > 4, S3 if n = 4.

Other names: PSO2n(R), projective special orthogonal group

Remarks: D3 is the same as A3, D2 is the same as A12, and D1 is abelian.

Dn I(n ≥ 4)

Dimension: n(2n − 1)

Real rank: min(p,q) (p+q=2n)

Maximal compact subgroup:

Fundamental group: Order 8 if p and q are both at least 3.

Outer automorphism group:

Other names: PSOp,q(R)

Dimension of symmetric space: pq

Compact symmetric space: Grassmannian of Rps in Rp+q. This is projective space if p or q is 1. Quaternion-Kähler if p or q is 4. Hermitian if p or q is 2.

Non-compact symmetric space: Grassmannian of positive definite Rps in Rp,q. This is hyperbolic space if p or q is 1. Quaternion-Kähler if p or q is 4. Hermitian if p or q is 2.

Remarks:

Dn III (n ≥ 4)

Dimension: n(2n − 1)

Real rank: n/2 or (n − 1)/2

Lie algebra of maximal compact subgroup: An−1R1

Fundamental group: Infinite `cyclic

Outer automorphism group: Order 2.

Other names:

Dimension of symmetric space: n(n − 1)

Compact symmetric space: Hermitian. Complex structures on R2n compatible with the Euclidean structure.

Non-compact symmetric space: Hermitian. Quaternionic quadratic forms on R2n.

Remarks:

Dn (n > 3) complex

Dimension: 2n(2n − 1)

Real rank: n

Maximal compact subgroup: Dn

Fundamental group: Order 4, (cyclic when n is odd).

Outer automorphism group: Noncyclic of order 4 for n>4, or the product of a group of order 2 and the symmetric group on 3 points when n=4.

Other names: Complex projective special orthogonal group, PSO2n(C)

Dimension of symmetric space: n(2n − 1)

Compact symmetric space: Compact group Dn

Non-compact symmetric space:

Remarks:

E6−78 (compact)

Dimension: 78

Real rank: 0

Fundamental group: 3

Outer automorphism group: 2

Other names:

Remarks:

E66 I (split)

Dimension: 78

Real rank: 6

Maximal compact subgroup: C4

Fundamental group: Order 2

Outer automorphism group: Order 2

Other names: E I

Dimension of symmetric space: 42

Compact symmetric space:

Non-compact symmetric space:

E62 II

Dimension: 78

Real rank: 4

Maximal compact subgroup: A5A1

Fundamental group: Cyclic, order 6.

Outer automorphism group: Order 2

Other names: E II

Dimension of symmetric space: 40

Compact symmetric space: Quaternion-Kähler.

Non-compact symmetric space: Quaternion-Kähler.

Remarks:

E6−14 III

Dimension: 78

Real rank: 2

Maximal compact subgroup: D5S1

Fundamental group: Infinite cyclic

Outer automorphism group: Trivial

Other names: E III

Dimension of symmetric space: 32

Compact symmetric space: Hermitian. Rosenfeld's elliptic projective plane over the complexified Cayley numbers.

Non-compact symmetric space: Hermitian. Rosenfeld's hyperbolic projective plane over the complexified Cayley numbers.

Remarks:

E6−26 IV

Dimension: 78

Real rank: 2

Maximal compact subgroup: F4

Fundamental group: Trivial

Outer automorphism group: Order 2

Other names: E IV

Dimension of symmetric space: 26

Compact symmetric space: Set of Cayley projective planes in the projective plane over the complexified Cayley numbers.

Non-compact symmetric space: Set of Cayley hyperbolic planes in the hyperbolic plane over the complexified Cayley numbers.

Remarks:

E6 complex

Dimension: 156

Real rank: 6

Maximal compact subgroup: E6

Fundamental group: 3

Outer automorphism group: Order 4 (non-cyclic)

Other names:

Dimension of symmetric space: 78

Compact symmetric space: Compact group E6

Non-compact symmetric space:

Remarks:

E7−133 (compact)

Dimension: 133

Real rank: 0

Fundamental group: 2

Outer automorphism group: 1

Other names:

Remarks:

E77 V (split)

Dimension: 133

Real rank: 7

Maximal compact subgroup: A7

Fundamental group: Cyclic, order 4

Outer automorphism group: Order 2

Other names:

Dimension of symmetric space: 70

Compact symmetric space:

Non-compact symmetric space:

Remarks:

E7−5 VI

Dimension: 133

Real rank: 4

Maximal compact subgroup: D6A1

Fundamental group: Non-cyclic, order 4

Outer automorphism group: Trivial

Other names:

Dimension of symmetric space: 64

Compact symmetric space: Quaternion-Kähler.

Non-compact symmetric space: Quaternion-Kähler.

Remarks:

E7−25 VII

Dimension: 133

Real rank: 3

Maximal compact subgroup: E6S1

Fundamental group: Infinite cyclic

Outer automorphism group: Order 2

Other names: E VII

Dimension of symmetric space: 54

Compact symmetric space:

Non-compact symmetric space:

Remarks: Symmetric spaces are Hermitian.

E7 complex

Dimension: 266

Real rank: 7

Maximal compact subgroup: E7

Fundamental group: 2

Outer automorphism group: Order 2 (complex conjugation)

Other names:

Dimension of symmetric space: 133

Compact symmetric space: Compact group E7

Non-compact symmetric space:

Remarks:

E8−248 compact

Dimension: 248

Real rank: 0

Fundamental group: 1

Outer automorphism group: 1

Other names:

Remarks:

E88 VIII (split)

Dimension: 248

Real rank: 8

Maximal compact subgroup: D8

Fundamental group: 2

Outer automorphism group: 1

Other names: E VIII

Dimension of symmetric space: 128

Compact symmetric space:

Non-compact symmetric space:

Remarks: @ E8

E8−24 IX

Dimension: 248

Real rank: 4

Maximal compact subgroup: E7 × A1

Fundamental group: Order 2.

Outer automorphism group: 1

Other names: E IX

Dimension of symmetric space: 112

Compact symmetric space: Quaternion-Kähler.

Non-compact symmetric space: Quaternion-Kähler.

Remarks:

E8 complex

Dimension: 496

Real rank: 8

Maximal compact subgroup: E8

Fundamental group: 1

Outer automorphism group: Order 2 (complex conjugation)

Other names:

Dimension of symmetric space: 248

Compact symmetric space: Compact group E8

Non-compact symmetric space:

Remarks:

F4−52 compact

Dimension: 52

Real rank: 0

Fundamental group: 1

Outer automorphism group: 1

Other names:

Remarks:

F44 I split

Dimension: 52

Real rank: 4

Maximal compact subgroup: C3 × A1

Fundamental group: Order 2

Outer automorphism group: 1

Other names: F I

Dimension of symmetric space: 28

Compact symmetric space: Quaternionic projective planes in Cayley projective plane.

Non-compact symmetric space: Hyperbolic quaternionic projective planes in hyperbolic Cayley projective plane.

Remarks:

F4−20 II

Dimension: 52

Real rank: 1

Maximal compact subgroup: B4 (Spin9(R))

Fundamental group: Order 2

Outer automorphism group: 1

Other names: F II

Dimension of symmetric space: 16

Compact symmetric space: Cayley projective plane. Quaternion-Kähler.

Non-compact symmetric space: Hyperbolic Cayley projective plane. Quaternion-Kähler.

Remarks:

F4 complex

Dimension: 104

Real rank: 4

Maximal compact subgroup: F4

Fundamental group: 1

Outer automorphism group: 2

Other names:

Dimension of symmetric space: 52

Compact symmetric space: Compact group F4

Non-compact symmetric space:

Remarks:

G2−14 compact

Dimension: 14

Real rank: 0

Fundamental group: 1

Outer automorphism group: 1

Other names:

Remarks: This is the automorphism group of the Cayley algebra.

G22 I split

Dimension: 14

Real rank: 2

Maximal compact subgroup: A1×A1

Fundamental group: Order 2

Outer automorphism group: 1

Other names: G I

Dimension of symmetric space: 8

Compact symmetric space: Quaternionic subalgebras of the Cayley algebra. Quaternion-Kähler.

Non-compact symmetric space: Non-division quaternionic subalgebras of the non-division Cayley algebra. Quaternion-Kähler.

Remarks:

G2 complex

Dimension: 28

Real rank: 2

Maximal compact subgroup: G2

Fundamental group: 1

Outer automorphism group: Order 2 (complex conjugation)

Other names:

Dimension of symmetric space: 14

Compact symmetric space: Compact group G2

Non-compact symmetric space:

Remarks:

Simple Lie groups of small dimension

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.

Dimension
1 R, S1=U(1)=SO2(R)
3 S3=Sp(1)=SU(2), SO3(R)=PSU(2) (Compact)
3 SL2(R)=Sp2(R), SO2,1(R)
6 SL2(C)=Sp2(C), SO3,1(R), SO3(C)
8 SL3(R'')
8 SU(3)
8 SU(1,2)
10 Sp(2), SO5(R)
10 SO4,1(R), Sp2,2(R)
10 SO3,2(R),Sp4(R)
14 G2 (Compact)
14 G2 (Split)
15 SU(4), SO6(R)
15 SL4(R), SO3,3(R)
15 SU(3,1)
15 SU(2,2), SO4,2(R)
15 SL2(H), SO5,1(R)
16 SL3(C)
20 SO5(C), Sp4(C)

Symmetric spaces of small dimension or rank

The non-compact simply connected irreducible symmetric spaces of rank 1 are given by hyperbolic spaces over the reals, complex numbers, quaternions, and the hyperbolic plane over the Cayley numbers. The compact duals are given by the corresponding projective spaces.

Here is a table of some simply connected irreducible symmetric spaces of small dimension:

Dimension Compact Non-compact
1 R
2 Sphere S2 Hyperbolic plane H2
3 Sphere S3 Hyperbolic space H3
4 Sphere S4 Hyperbolic space H4
4 Complex projective space CP2 Complex hyperbolic plane CH2

Further reading

  • Besse, Einstein manifolds ISBN 0-387-15279-2
  • Helgason, Differential geometry, Lie groups, and symmetric spaces. ISBN 0-8218-2848-7
  • Füchs and Schweigert, Symmetries, Lie algebras, and representations: a graduate course for physicists. Cambridge University Press, 2003. ISBN 0-521-54119-0
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