List of simple Lie groups
Encyclopedia
In mathematics
, the simple Lie group
s 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 space
s. 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.
, 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.
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 (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.
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.
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.
s.
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).
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).
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
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:
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:
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:
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:
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.
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:
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:
Real rank: 0
Fundamental group: 2
Outer automorphism group: 1
Other names: Sp(n), Sp(2n), USp(n), USp(2n), compact symplectic group
Remarks: C1 is the same as B1
and A1. C2 is the same as B2.
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. Complex structures on R2n compatible with a symplectic form. 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.
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:
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:
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.
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:
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:
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:
Real rank: 0
Fundamental group: 3
Outer automorphism group: 2
Other names:
Remarks:
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:
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:
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:
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:
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:
Real rank: 0
Fundamental group: 2
Outer automorphism group: 1
Other names:
Remarks:
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:
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:
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.
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:
Real rank: 0
Fundamental group: 1
Outer automorphism group: 1
Other names:
Remarks:
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
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:
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:
Real rank: 0
Fundamental group: 1
Outer automorphism group: 1
Other names:
Remarks:
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:
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:
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:
Real rank: 0
Fundamental group: 1
Outer automorphism group: 1
Other names:
Remarks:
This is the automorphism group of the Cayley algebra.
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:
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:
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.
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:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the simple Lie group
Simple Lie group
In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...
s were classified by Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
.
The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric space
Riemannian symmetric space
In differential geometry, representation theory and harmonic analysis, a symmetric space is a smooth manifold whose group of symmetries contains an inversion symmetry about every point. There are two ways to formulate the inversion symmetry, via Riemannian geometry or via Lie theory...
s. See also the table of Lie groups
Table of Lie groups
This article gives a table of some common Lie groups and their associated Lie algebras.The following are noted: the topological properties of the group , as well as on their algebraic properties .For more examples of Lie groups and other...
for a smaller list of groups that commonly occur in theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, and the Bianchi classification
Bianchi classification
In mathematics, the Bianchi classification, named for Luigi Bianchi, is a classification of the 3-dimensional real Lie algebras into 11 classes, 9 of which are single groups and two of which have a continuum of isomorphism classes...
for groups of dimension at most 3.
Simple Lie groups
Unfortunately there is no generally accepted definition of a simple Lie groupSimple Lie group
In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...
, and in particular it is not necessarily defined as a Lie group that is simple
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
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, andthis 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 (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
R, C, H, and O stand for the real numbers, complex numbers, quaternions, and octonionOctonion
In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...
s.
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).
R (Abelian)
Dimension: 1Outer 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
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
.
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 ≤ p ≤ q)
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
Hermitian symmetric space
In mathematics, a Hermitian symmetric space is a Kähler manifold M which, as a Riemannian manifold, is a Riemannian symmetric space. Equivalently, M is a Riemannian symmetric space with a parallel complex structure with respect to which the Riemannian metric is 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
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
.
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), compact symplectic group
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. Complex structures on R2n compatible with a symplectic form. 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 ≤ p ≤ q)
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
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
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: 78Real rank: 0
Fundamental group: 3
Outer automorphism group: 2
Other names:
Remarks:
E66 I (split)
Dimension: 78Real 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: 78Real 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: 78Real 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: 78Real 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: 156Real 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: 133Real rank: 0
Fundamental group: 2
Outer automorphism group: 1
Other names:
Remarks:
E77 V (split)
Dimension: 133Real 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: 133Real 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: 133Real 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: 266Real 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: 248Real rank: 0
Fundamental group: 1
Outer automorphism group: 1
Other names:
Remarks:
E88 VIII (split)
Dimension: 248Real 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: 248Real 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: 496Real 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: 52Real rank: 0
Fundamental group: 1
Outer automorphism group: 1
Other names:
Remarks:
F44 I split
Dimension: 52Real 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: 52Real 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: 104Real 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: 14Real 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: 14Real 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: 28Real 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 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.
| Dimension | |
|---|---|
| 1 | R, S1=U(1)=SO2(R)=Spin(2) |
| 3 | S3=Sp(1)=SU(2)=Spin(3), 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)=Spin(5), SO5(R) |
| 10 | SO4,1(R), Sp2,2(R) |
| 10 | SO3,2(R),Sp4(R) |
| 14 | G2 (Compact) |
| 14 | G2 (Split) |
| 15 | SU(4)=Spin(6), 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 givenby 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