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.

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[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 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[edit]

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.

Related ideas[edit]

Simple Lie algebras[edit]

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[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 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[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 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[edit]

  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).

Full classification[edit]

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[edit]

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).

Dynkin diagrams

For the infinite (A, B, C, D) series of Dynkin diagrams, the simply 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.

Overview of the classification[edit]

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[edit]

Abelian[edit]

Dimension Outer automorphism group Dimension of symmetric space Symmetric space Remarks
(Abelian) 1 1

Notes[edit]

^† 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[edit]

Dimension Real rank Fundamental
group
Outer automorphism
group
Other names Remarks
An (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)
A1 is the same as B1 and C1
Bn (n ≥ 2) compact n(2n + 1) 0 2 1 special orthogonal group
SO2n+1(R)
B1 is the same as A1 and C1.
B2 is the same as C2.
Cn (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 Hn. Copies of complex projective space in quaternionic projective space.
Dn (n ≥ 4) compact n(2n − 1) 0 Order 4 (cyclic when n is odd). 2 if n > 4, S3 if n = 4 projective special orthogonal group
PSO2n(R)
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[edit]

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 (n ≥ 1) split n(n + 2) n Dn/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
PSLn+1(R)
n(n + 3)/2 Real 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) split n(2n + 1) n SO(n)SO(n+1) Non-cyclic, order 4 1 identity component of special orthogonal group
SO(n,n+1)
n(n + 1) B1 is the same as A1.
Cn I (n ≥ 3) split n(2n + 1) n An−1S1 Infinite cyclic 1 projective 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) split n(2n - 1) n SO(n)SO(n) Order 4 if n odd, 8 if n even 2 if n > 4, S3 if n = 4 identity component of projective special orthogonal group
PSO(n,n)
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[edit]

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 (n ≥ 1) complex 2n(n + 2) n An Cyclic, order n + 1 2 if n = 1, 4 (noncyclic) if n ≥ 2. projective complex special linear group
PSLn+1(C)
n(n + 2) Compact group An Hermitian forms on Cn+1

with fixed volume.

Bn (n ≥ 2) complex 2n(2n + 1) n Bn 2 Order 2 (complex conjugation) complex special orthogonal group
SO2n+1(C)
n(2n + 1) Compact group Bn
Cn (n ≥ 3) complex 2n(2n + 1) n Cn 2 Order 2 (complex conjugation) projective complex symplectic group
PSp2n(C)
n(2n + 1) Compact group Cn
Dn (n ≥ 4) complex 2n(2n − 1) n Dn 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 S3 when n = 4. projective complex special orthogonal group
PSO2n(C)
n(2n − 1) Compact group Dn
E6 complex 156 6 E6 3 Order 4 (non-cyclic) 78 Compact group E6
E7 complex 266 7 E7 2 Order 2 (complex conjugation) 133 Compact group E7
E8 complex 496 8 E8 1 Order 2 (complex conjugation) 248 Compact group E8
F4 complex 104 4 F4 1 2 52 Compact group F4
G2 complex 28 2 G2 1 Order 2 (complex conjugation) 14 Compact group G2

Others[edit]

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 ≥ 2)
(2n − 1)(2n + 1) n − 1 Cn Order 2 SLn(H), SU(2n) (n − 1)(2n + 1) Quaternionic structures on C2n compatible with the Hermitian structure Copies 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) p Ap−1Aq−1S1 SU(p,q), 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 |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) 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 |pq| ≤ 1, split.
Cn II
(n > 2)
n = p+q
(1 ≤ pq)
n(2n + 1) min(p,q) CpCq Order 2 1 if pq, 2 if p = q. Sp2p,2q(R) 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
(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 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⌋ An−1R1 Infinite cyclic Order 2 SO*(2n) n(n − 1) Hermitian.
Complex structures on R2n compatible with the Euclidean structure.
Hermitian.
Quaternionic quadratic forms on R2n.
E62 II
(quasi-split)
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 (Spin9(R)) Order 2 1 F II 16 Cayley projective plane. Quaternion-Kähler. Hyperbolic Cayley projective plane. Quaternion-Kähler.

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 , S1 = U(1) = SO2() = Spin(2) Abelian Real line 0 1
3 S3 = Sp(1) = SU(2)=Spin(3), SO3() = PSU(2) Compact
3 SL2() = Sp2(), SO2,1() Split, Hermitian, hyperbolic Hyperbolic plane Sphere S2 1 2
6 SL2() = Sp2(), SO3,1(), SO3() Complex Hyperbolic space Sphere S3 1 3
8 SL3() Split Euclidean structures on Real structures on 2 5
8 SU(3) Compact
8 SU(1,2) Hermitian, quasi-split, quaternionic Complex hyperbolic plane Complex projective plane 1 4
10 Sp(2) = Spin(5), SO5() Compact
10 SO4,1(), Sp2,2() Hyperbolic, quaternionic Hyperbolic space Sphere S4 1 4
10 SO3,2(), Sp4() Split, Hermitian Siegel upper half space Complex structures on 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(4) = Spin(6), SO6() Compact
15 SL4(), SO3,3() 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), SO4,2() Hermitian, quasi-split, quaternionic 2 in 2,4 Grassmannian G(2,4) 2 8
15 SL2(), SO5,1() Hyperbolic Hyperbolic space Sphere S5 1 5
16 SL3() Complex SU(3) 2 8
20 SO5(), Sp4() Complex Spin5() 2 10
21 SO7() Compact
21 SO6,1() Hyperbolic Hyperbolic space Sphere S6
21 SO5,2() Hermitian
21 SO4,3() Split, quaternionic
21 Sp(3) Compact
21 Sp6() Split, hermitian
21 Sp4,2() Quaternionic
24 SU(5) Compact
24 SL5() Split
24 SU4,1 Hermitian
24 SU3,2 Hermitian, quaternionic
28 SO8() Compact
28 SO7,1() Hyperbolic Hyperbolic space Sphere S7
28 SO6,2() Hermitian
28 SO5,3() Quasi-split
28 SO4,4() Split, quaternionic
28 SO8() Hermitian
28 G2() Complex
30 SL4() Complex

Simply laced groups[edit]

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[edit]

References[edit]

  • Jacobson, Nathan (1971). Exceptional Lie Algebras. CRC Press. ISBN 0-8247-1326-5.
  • Fulton, William; Harris, Joe (2004). Representation Theory: A First Course. Springer. doi:10.1007/978-1-4612-0979-9. ISBN 978-1-4612-0979-9.

Further reading[edit]

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