Simple Lie group

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In mathematics, a simple Lie group is a connected non-abelian 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.

Contents

Together with the commutative Lie group of the real numbers, , and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "blocks" that make up all (finite-dimensional) 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 so-called "special linear group" SL(n) of n by n matrices with determinant equal to 1 is simple for all n > 1.

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.

Definition

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 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 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.

Alternatives

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, hence being a simple Lie group is different from being simple as an abstract group.

Simple Lie groups include many classical Lie groups, which provide a group-theoretic 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 counter-example are the special orthogonal groups in even dimension. These have the matrix in the center, and this element is path-connected to the identity element, and so these groups evade the definition. Both of these are reductive groups.

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. (Authors differ on whether the one-dimensional 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

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

  stand for the real numbers, complex numbers, quaternions, and octonions.

In the symbols such as E626 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).

Full classification

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 infinite-dimensional 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

Compact Lie groups

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).

Finite Dynkin diagrams.svg

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.

Overview of the classification

Ar 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).

Br 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.

Cr 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 double-cover by the metaplectic group.

Dr 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 D2 is two isolated nodes, the same as A1 A1, 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 D3 is the same as A3, corresponding to a covering map homomorphism from SU(4) to SO(6).

In addition to the four families Ai, Bi, Ci, and Di above, there are five so-called exceptional Dynkin diagrams G2, F4, E6, E7, and E8; 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 G2 is the automorphism group of the octonions, and the group associated to F4 is the automorphism group of a certain Albert algebra.

See also E7+12.

List

Abelian

DimensionOuter automorphism groupDimension of symmetric spaceSymmetric spaceRemarks
(Abelian)11

Notes

^† The group is not 'simple' as an abstract group, and according to most (but not all) definitions this is not a simple Lie group. Further, 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 is the only such non-compact symmetric space without a compact dual (although it has a compact quotient S1).

Compact

DimensionReal rankFundamental
group
Outer automorphism
group
Other namesRemarks
An (n ≥ 1) compactn(n + 2)0Cyclic, order n + 11 if n = 1, 2 if n > 1. projective special unitary group
PSU(n + 1)
A1 is the same as B1 and C1
Bn (n ≥ 2) compactn(2n + 1)021 special orthogonal group
SO2n+1(R)
B1 is the same as A1 and C1.
B2 is the same as C2.
Cn (n ≥ 3) compactn(2n + 1)021projective compact symplectic group
PSp(n), PSp(2n), PUSp(n), PUSp(2n)
Hermitian. Complex structures of Hn. Copies of complex projective space in quaternionic projective space.
Dn (n ≥ 4) compactn(2n 1)0Order 4 (cyclic when n is odd).2 if n > 4, S3 if n = 4projective special orthogonal group
PSO2n(R)
D3 is the same as A3, D2 is the same as A12, and D1 is abelian.
E678 compact78032
E7133 compact133021
E8248 compact248011
F452 compact52011
G214 compact14011This is the automorphism group of the Cayley algebra.

Split

DimensionReal rankMaximal compact
subgroup
Fundamental
group
Outer automorphism
group
Other namesDimension of
symmetric space
Compact
symmetric space
Non-Compact
symmetric space
Remarks
An I (n ≥ 1) splitn(n + 2)nDn/2 or B(n1)/2Infinite cyclic if n = 1
2 if n ≥ 2
1 if n = 1
2 if n ≥ 2.
projective special linear group
PSLn+1(R)
n(n + 3)/2Real structures on Cn+1 or set of RPn in CPn. Hermitian if n = 1, in which case it is the 2-sphere.Euclidean structures on Rn+1. Hermitian if n = 1, when it is the upper half plane or unit complex disc.
Bn I (n ≥ 2) splitn(2n + 1)nSO(n)SO(n+1)Non-cyclic, order 41identity component of special orthogonal group
SO(n,n+1)
n(n + 1)B1 is the same as A1.
Cn I (n ≥ 3) splitn(2n + 1)nAn1S1Infinite cyclic1projective symplectic group
PSp2n(R), PSp(2n,R), PSp(2n), PSp(n,R), PSp(n)
n(n + 1)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 (n ≥ 4) splitn(2n - 1)nSO(n)SO(n)Order 4 if n odd, 8 if n even2 if n > 4, S3 if n = 4identity component of projective special orthogonal group
PSO(n,n)
n2D3 is the same as A3, D2 is the same as A12, and D1 is abelian.
E66 I split786C4Order 2Order 2E I42
E77 V split1337A7Cyclic, order 4Order 270
E88 VIII split2488D821E VIII128@ E8
F44 I split524C3 × A1Order 21F I28Quaternionic projective planes in Cayley projective plane.Hyperbolic quaternionic projective planes in hyperbolic Cayley projective plane.
G22 I split142A1 × A1Order 21G I8Quaternionic subalgebras of the Cayley algebra. Quaternion-Kähler.Non-division quaternionic subalgebras of the non-division Cayley algebra. Quaternion-Kähler.

Complex

Real dimensionReal rankMaximal compact
subgroup
Fundamental
group
Outer automorphism
group
Other namesDimension of
symmetric space
Compact
symmetric space
Non-Compact
symmetric space
An (n ≥ 1) complex2n(n + 2)nAnCyclic, order n + 12 if n = 1, 4 (noncyclic) if n ≥ 2.projective complex special linear group
PSLn+1(C)
n(n + 2)Compact group AnHermitian forms on Cn+1

with fixed volume.

Bn (n ≥ 2) complex2n(2n + 1)nBn2Order 2 (complex conjugation)complex special orthogonal group
SO2n+1(C)
n(2n + 1)Compact group Bn
Cn (n ≥ 3) complex2n(2n + 1)nCn2Order 2 (complex conjugation)projective complex symplectic group
PSp2n(C)
n(2n + 1)Compact group Cn
Dn (n ≥ 4) complex2n(2n 1)nDnOrder 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 S3 when n = 4.projective complex special orthogonal group
PSO2n(C)
n(2n 1)Compact group Dn
E6 complex1566E63Order 4 (non-cyclic)78Compact group E6
E7 complex2667E72Order 2 (complex conjugation)133Compact group E7
E8 complex4968E81Order 2 (complex conjugation)248Compact group E8
F4 complex1044F41252Compact group F4
G2 complex282G21Order 2 (complex conjugation)14Compact group G2

Others

DimensionReal rankMaximal compact
subgroup
Fundamental
group
Outer automorphism
group
Other namesDimension of
symmetric space
Compact
symmetric space
Non-Compact
symmetric space
Remarks
A2n1 II
(n ≥ 2)
(2n 1)(2n + 1)n 1CnOrder 2SLn(H), SU(2n)Quaternionic structures on C2n compatible with the Hermitian structureCopies of quaternionic hyperbolic space (of dimension n 1) in complex hyperbolic space (of dimension 2n 1).
An III
(n ≥ 1)
p + q = n + 1
(1 ≤ pq)
n(n + 2)pAp1Aq1S1SU(p,q), A III2pq 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 |pq| ≤ 1, quasi-split
Bn I
(n > 1)
p+q = 2n+1
n(2n + 1)min(p,q)SO(p)SO(q) SO(p,q) pqGrassmannian 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 |pq| ≤ 1, split.
Cn II
(n > 2)
n = p+q
(1 ≤ pq)
n(2n + 1)min(p,q)CpCqOrder 21 if pq, 2 if p = q.Sp2p,2q(R)4pqGrassmannian 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
(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)pqGrassmannian 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 p = q, split
If |pq| ≤ 2, quasi-split
Dn III
(n ≥ 4)
n(2n 1)n/2⌋An1R1Infinite cyclicOrder 2SO*(2n)n(n 1)Hermitian.
Complex structures on R2n compatible with the Euclidean structure.
Hermitian.
Quaternionic quadratic forms on R2n.
E62 II
(quasi-split)
784A5A1Cyclic, order 6Order 2E II40Quaternion-Kähler.Quaternion-Kähler.Quasi-split but not split.
E614 III782D5S1Infinite cyclicTrivialE III32Hermitian.
Rosenfeld elliptic projective plane over the complexified Cayley numbers.
Hermitian.
Rosenfeld hyperbolic projective plane over the complexified Cayley numbers.
E626 IV782F4TrivialOrder 2E IV26Set 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.
E75 VI1334D6A1Non-cyclic, order 4TrivialE VI64Quaternion-Kähler.Quaternion-Kähler.
E725 VII1333E6S1Infinite cyclicOrder 2E VII54Hermitian.Hermitian.
E824 IX2484E7 × A1Order 21E IX112Quaternion-Kähler.Quaternion-Kähler.
F420 II521B4 (Spin9(R))Order 21F II16Cayley 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 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.

DimGroupsSymmetric spaceCompact dualRankDim
1, S1 = U(1) = SO2() = Spin(2)AbelianReal line01
3S3 = Sp(1) = SU(2)=Spin(3), SO3() = PSU(2)Compact
3SL2() = Sp2(), SO2,1()Split, Hermitian, hyperbolicHyperbolic plane Sphere S212
6SL2() = Sp2(), SO3,1(), SO3()ComplexHyperbolic space Sphere S313
8SL3()SplitEuclidean structures on Real structures on 25
8SU(3)Compact
8SU(1,2)Hermitian, quasi-split, quaternionicComplex hyperbolic planeComplex projective plane14
10Sp(2) = Spin(5), SO5()Compact
10SO4,1(), Sp2,2()Hyperbolic, quaternionicHyperbolic space Sphere S414
10SO3,2(), Sp4()Split, HermitianSiegel upper half spaceComplex structures on 26
14G2Compact
14G2Split, quaternionicNon-division quaternionic subalgebras of non-division octonionsQuaternionic subalgebras of octonions28
15SU(4) = Spin(6), SO6()Compact
15SL4(), SO3,3()Split3 in 3,3Grassmannian G(3,3)39
15SU(3,1)HermitianComplex hyperbolic spaceComplex projective space16
15SU(2,2), SO4,2()Hermitian, quasi-split, quaternionic2 in 2,4Grassmannian G(2,4)28
15SL2(), SO5,1()HyperbolicHyperbolic space Sphere S515
16SL3()ComplexSU(3)28
20SO5(), Sp4()ComplexSpin5()210
21SO7()Compact
21SO6,1()HyperbolicHyperbolic space Sphere S6
21SO5,2()Hermitian
21SO4,3()Split, quaternionic
21Sp(3)Compact
21Sp6()Split, hermitian
21Sp4,2()Quaternionic
24SU(5)Compact
24SL5()Split
24SU4,1Hermitian
24SU3,2Hermitian, quaternionic
28SO8()Compact
28SO7,1()HyperbolicHyperbolic space Sphere S7
28SO6,2()Hermitian
28SO5,3()Quasi-split
28SO4,4()Split, quaternionic
28SO8()Hermitian
28G2()Complex
30SL4()Complex

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, but no group of type B, C, F, or G is simply laced.

See also

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References

    Further reading