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Lie groups and Lie algebras |
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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.

- Definition
- Alternatives
- Related ideas
- Simple Lie algebras
- Symmetric spaces
- Hermitian symmetric spaces
- Notation
- Full classification
- Compact Lie groups
- Overview of the classification
- List
- Abelian
- Compact
- Split
- Complex
- Others
- Simple Lie groups of small dimension
- Simply laced groups
- See also
- References
- Further reading

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 first classification of simple Lie groups was by Wilhelm Killing, and this work was later perfected by Élie Cartan. The final classification is often referred to as Killing-Cartan classification.

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.

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. For this reason, the definition of a simple Lie group is not equivalent to the definition of a Lie group that is 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.

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 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.

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.

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

In the symbols such as *E*_{6}^{−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).

Simple Lie groups are fully classified. The classification is usually stated in several steps, namely:

- Classification of simple complex Lie algebras The classification of simple Lie algebras over the complex numbers by Dynkin diagrams.
- Classification of simple real Lie algebras Each simple complex Lie algebra has several real forms, classified by additional decorations of its Dynkin diagram called Satake diagrams, after Ichirô Satake.
**Classification of centerless simple Lie groups**For every (real or complex) simple Lie algebra , there is a unique "centerless" simple Lie group whose Lie algebra is and which has trivial center.- Classification of simple Lie groups

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

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).

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.

A_{r} 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).

B_{r} has as its associated centerless compact groups the odd special orthogonal groups, SO(2*r* + 1). This group is not simply connected however: its universal (double) cover is the spin group.

C_{r} 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.

D_{r} has as its associated compact group the even special orthogonal groups, SO(2*r*) and as its associated centerless compact group the projective special orthogonal group PSO(2*r*) = SO(2*r*)/{I, −I}. As with the B series, SO(2*r*) is not simply connected; its universal cover is again the spin group, but the latter again has a center (cf. its article).

The diagram D_{2} is two isolated nodes, the same as A_{1}∪ A_{1}, 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 D_{3} is the same as A_{3}, corresponding to a covering map homomorphism from SU(4) to SO(6).

In addition to the four families *A*_{i}, *B*_{i}, *C*_{i}, and *D*_{i} above, there are five so-called exceptional Dynkin diagrams G_{2}, F_{4}, E_{6}, E_{7}, and E_{8}; 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 G_{2} is the automorphism group of the octonions, and the group associated to F_{4} is the automorphism group of a certain Albert algebra.

Dimension | Outer automorphism group | Dimension of symmetric space | Symmetric space | Remarks | |
---|---|---|---|---|---|

(Abelian) | 1 | 1 | ^{ † } |

**^†**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*S*^{1}).

Dimension | Real rank | Fundamental group | Outer automorphism group | Other names | Remarks | |
---|---|---|---|---|---|---|

A_{n} (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) | A_{1} is the same as B_{1} and C_{1} |

B_{n} (n ≥ 2) compact | n(2n + 1) | 0 | 2 | 1 | special orthogonal group SO _{2n+1}(R) | B_{1} is the same as A_{1} and C_{1}.B_{2} is the same as C_{2}. |

C_{n} (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 H^{n}. Copies of complex projective space in quaternionic projective space. |

D_{n} (n ≥ 4) compact | n(2n− 1) | 0 | Order 4 (cyclic when n is odd). | 2 if n > 4, S_{3} if n = 4 | projective special orthogonal group PSO _{2n}(R) | D_{3} is the same as A_{3}, D_{2} is the same as A_{1}^{2}, and D_{1} is abelian. |

E_{6}^{−78} compact | 78 | 0 | 3 | 2 | ||

E_{7}^{−133} compact | 133 | 0 | 2 | 1 | ||

E_{8}^{−248} compact | 248 | 0 | 1 | 1 | ||

F_{4}^{−52} compact | 52 | 0 | 1 | 1 | ||

G_{2}^{−14} compact | 14 | 0 | 1 | 1 | This is the automorphism group of the Cayley algebra. |

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 | |
---|---|---|---|---|---|---|---|---|---|---|

A_{n} I (n ≥ 1) split | n(n + 2) | n | D_{n/2} or B_{(n−1)/2} | Infinite cyclic if n = 12 if n ≥ 2 | 1 if n = 12 if n ≥ 2. | projective special linear group PSL _{n+1}(R) | n(n + 3)/2 | Real structures on C^{n+1} or set of RP^{n} in CP^{n}. Hermitian if n = 1, in which case it is the 2-sphere. | Euclidean structures on R^{n+1}. Hermitian if n = 1, when it is the upper half plane or unit complex disc. | |

B_{n} 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) | B_{1} is the same as A_{1}. | ||

C_{n} I (n ≥ 3) split | n(2n + 1) | n | A_{n−1}S^{1} | Infinite cyclic | 1 | projective symplectic group PSp _{2n}(R), PSp(2n,R), PSp(2n), PSp(n,R), PSp(n) | n(n + 1) | Hermitian. Complex structures of H^{n}. Copies of complex projective space in quaternionic projective space. | Hermitian. Complex structures on R^{2n} compatible with a symplectic form. Set of complex hyperbolic spaces in quaternionic hyperbolic space. Siegel upper half space. | C_{2} is the same as B_{2}, and C_{1} is the same as B_{1} and A_{1}. |

D_{n} 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, S_{3} if n = 4 | identity component of projective special orthogonal group PSO( n,n) | n^{2} | D_{3} is the same as A_{3}, D_{2} is the same as A_{1}^{2}, and D_{1} is abelian. | ||

E_{6}^{6} I split | 78 | 6 | C_{4} | Order 2 | Order 2 | E I | 42 | |||

E_{7}^{7} V split | 133 | 7 | A_{7} | Cyclic, order 4 | Order 2 | 70 | ||||

E_{8}^{8} VIII split | 248 | 8 | D_{8} | 2 | 1 | E VIII | 128 | @ E8 | ||

F_{4}^{4} I split | 52 | 4 | C_{3} × A_{1} | Order 2 | 1 | F I | 28 | Quaternionic projective planes in Cayley projective plane. | Hyperbolic quaternionic projective planes in hyperbolic Cayley projective plane. | |

G_{2}^{2} I split | 14 | 2 | A_{1} × A_{1} | 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. |

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 | |
---|---|---|---|---|---|---|---|---|---|

A_{n} (n ≥ 1) complex | 2n(n + 2) | n | A_{n} | Cyclic, order n + 1 | 2 if n = 1, 4 (noncyclic) if n ≥ 2. | projective complex special linear group PSL _{n+1}(C) | n(n + 2) | Compact group A_{n} | Hermitian forms on C^{n+1}with fixed volume. |

B_{n} (n ≥ 2) complex | 2n(2n + 1) | n | B_{n} | 2 | Order 2 (complex conjugation) | complex special orthogonal group SO _{2n+1}(C) | n(2n + 1) | Compact group B_{n} | |

C_{n} (n ≥ 3) complex | 2n(2n + 1) | n | C_{n} | 2 | Order 2 (complex conjugation) | projective complex symplectic group PSp _{2n}(C) | n(2n + 1) | Compact group C_{n} | |

D_{n} (n ≥ 4) complex | 2n(2n− 1) | n | D_{n} | 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 S_{3} when n = 4. | projective complex special orthogonal groupPSO _{2n}(C) | n(2n− 1) | Compact group D_{n} | |

E_{6} complex | 156 | 6 | E_{6} | 3 | Order 4 (non-cyclic) | 78 | Compact group E_{6} | ||

E_{7} complex | 266 | 7 | E_{7} | 2 | Order 2 (complex conjugation) | 133 | Compact group E_{7} | ||

E_{8} complex | 496 | 8 | E_{8} | 1 | Order 2 (complex conjugation) | 248 | Compact group E_{8} | ||

F_{4} complex | 104 | 4 | F_{4} | 1 | 2 | 52 | Compact group F_{4} | ||

G_{2} complex | 28 | 2 | G_{2} | 1 | Order 2 (complex conjugation) | 14 | Compact group G_{2} |

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 | |
---|---|---|---|---|---|---|---|---|---|---|

A_{2n−1} II( n ≥ 2) | (2n− 1)(2n + 1) | n− 1 | C_{n} | Order 2 | SL_{n}(H), SU^{∗}(2n) | Quaternionic structures on C^{2n} compatible with the Hermitian structure | Copies of quaternionic hyperbolic space (of dimension n− 1) in complex hyperbolic space (of dimension 2n− 1). | |||

A_{n} III( n ≥ 1)p + q = n + 1(1 ≤ p ≤ q) | n(n + 2) | p | A_{p−1}A_{q−1}S^{1} | SU(p,q), A III | 2pq | Hermitian. Grassmannian of p subspaces of C^{p+q}.If p or q is 2; quaternion-Kähler | Hermitian. Grassmannian of maximal positive definite subspaces of C^{p,q}.If p or q is 2, quaternion-Kähler | If p=q=1, splitIf | p−q| ≤ 1, quasi-split | ||

B_{n} I( n > 1)p+q = 2n+1 | n(2n + 1) | min(p,q) | SO(p)SO(q) | SO(p,q) | pq | Grassmannian of R^{p}s in R^{p+q}.If p or q is 1, Projective spaceIf p or q is 2; HermitianIf p or q is 4, quaternion-Kähler | Grassmannian of positive definite R^{p}s in R^{p,q}.If p or q is 1, Hyperbolic spaceIf p or q is 2, HermitianIf p or q is 4, quaternion-Kähler | If |p−q| ≤ 1, split. | ||

C_{n} II( n > 2)n = p+q(1 ≤ p ≤ q) | n(2n + 1) | min(p,q) | C_{p}C_{q} | Order 2 | 1 if p ≠ q, 2 if p = q. | Sp_{2p,2q}(R) | 4pq | Grassmannian of H^{p}s in H^{p+q}.If p or q is 1, quaternionic projective spacein which case it is quaternion-Kähler. | H^{p}s in H^{p,q}.If p or q is 1, quaternionic hyperbolic spacein which case it is quaternion-Kähler. | |

D_{n} 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 R^{p}s in R^{p+q}.If p or q is 1, Projective spaceIf p or q is 2 ; HermitianIf p or q is 4, quaternion-Kähler | Grassmannian of positive definite R^{p}s in R^{p,q}.If p or q is 1, Hyperbolic SpaceIf p or q is 2, HermitianIf p or q is 4, quaternion-Kähler | If p = q, splitIf | p−q| ≤ 2, quasi-split | |

D_{n} III( n ≥ 4) | n(2n− 1) | ⌊n/2⌋ | A_{n−1}R^{1} | Infinite cyclic | Order 2 | SO^{*}(2n) | n(n− 1) | Hermitian. Complex structures on R ^{2n} compatible with the Euclidean structure. | Hermitian. Quaternionic quadratic forms on R ^{2n}. | |

E_{6}^{2} II(quasi-split) | 78 | 4 | A_{5}A_{1} | Cyclic, order 6 | Order 2 | E II | 40 | Quaternion-Kähler. | Quaternion-Kähler. | Quasi-split but not split. |

E_{6}^{−14} III | 78 | 2 | D_{5}S^{1} | 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. | |

E_{6}^{−26} IV | 78 | 2 | F_{4} | 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. | |

E_{7}^{−5} VI | 133 | 4 | D_{6}A_{1} | Non-cyclic, order 4 | Trivial | E VI | 64 | Quaternion-Kähler. | Quaternion-Kähler. | |

E_{7}^{−25} VII | 133 | 3 | E_{6}S^{1} | Infinite cyclic | Order 2 | E VII | 54 | Hermitian. | Hermitian. | |

E_{8}^{−24} IX | 248 | 4 | E_{7} × A_{1} | Order 2 | 1 | E IX | 112 | Quaternion-Kähler. | Quaternion-Kähler. | |

F_{4}^{−20} II | 52 | 1 | B_{4} (Spin_{9}(R)) | Order 2 | 1 | F II | 16 | Cayley projective plane. Quaternion-Kähler. | Hyperbolic Cayley projective plane. Quaternion-Kähler. |

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 | ℝ, S^{1} = U(1) = SO_{2}(ℝ) = Spin(2) | Abelian | Real line | 0 | 1 | |

3 | S^{3} = Sp(1) = SU(2)=Spin(3), SO_{3}(ℝ) = PSU(2) | Compact | ||||

3 | SL_{2}(ℝ) = Sp_{2}(ℝ), SO_{2,1}(ℝ) | Split, Hermitian, hyperbolic | Hyperbolic plane | Sphere S^{2} | 1 | 2 |

6 | SL_{2}(ℂ) = Sp_{2}(ℂ), SO_{3,1}(ℝ), SO_{3}(ℂ) | Complex | Hyperbolic space | Sphere S^{3} | 1 | 3 |

8 | SL_{3}(ℝ) | 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), SO_{5}(ℝ) | Compact | ||||

10 | SO_{4,1}(ℝ), Sp_{2,2}(ℝ) | Hyperbolic, quaternionic | Hyperbolic space | Sphere S^{4} | 1 | 4 |

10 | SO_{3,2}(ℝ), Sp_{4}(ℝ) | Split, Hermitian | Siegel upper half space | Complex structures on | 2 | 6 |

14 | G_{2} | Compact | ||||

14 | G_{2} | Split, quaternionic | Non-division quaternionic subalgebras of non-division octonions | Quaternionic subalgebras of octonions | 2 | 8 |

15 | SU(4) = Spin(6), SO_{6}(ℝ) | Compact | ||||

15 | SL_{4}(ℝ), SO_{3,3}(ℝ) | Split | ℝ^{3} in ℝ^{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), SO_{4,2}(ℝ) | Hermitian, quasi-split, quaternionic | ℝ^{2} in ℝ^{2,4} | Grassmannian G(2,4) | 2 | 8 |

15 | SL_{2}(ℍ), SO_{5,1}(ℝ) | Hyperbolic | Hyperbolic space | Sphere S^{5} | 1 | 5 |

16 | SL_{3}(ℂ) | Complex | SU(3) | 2 | 8 | |

20 | SO_{5}(ℂ), Sp_{4}(ℂ) | Complex | Spin_{5}(ℝ) | 2 | 10 | |

21 | SO_{7}(ℝ) | Compact | ||||

21 | SO_{6,1}(ℝ) | Hyperbolic | Hyperbolic space | Sphere S^{6} | ||

21 | SO_{5,2}(ℝ) | Hermitian | ||||

21 | SO_{4,3}(ℝ) | Split, quaternionic | ||||

21 | Sp(3) | Compact | ||||

21 | Sp_{6}(ℝ) | Split, hermitian | ||||

21 | Sp_{4,2}(ℝ) | Quaternionic | ||||

24 | SU(5) | Compact | ||||

24 | SL_{5}(ℝ) | Split | ||||

24 | SU_{4,1} | Hermitian | ||||

24 | SU_{3,2} | Hermitian, quaternionic | ||||

28 | SO_{8}(ℝ) | Compact | ||||

28 | SO_{7,1}(ℝ) | Hyperbolic | Hyperbolic space | Sphere S^{7} | ||

28 | SO_{6,2}(ℝ) | Hermitian | ||||

28 | SO_{5,3}(ℝ) | Quasi-split | ||||

28 | SO_{4,4}(ℝ) | Split, quaternionic | ||||

28 | SO^{∗}_{8}(ℝ) | Hermitian | ||||

28 | G_{2}(ℂ) | Complex | ||||

30 | SL_{4}(ℂ) | Complex |

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.

In mathematics, a **Lie group** is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.

In mathematics, a **root system** is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory. Finally, root systems are important for their own sake, as in spectral graph theory.

In the mathematical field of Lie theory, a **Dynkin diagram**, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled. Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various properties of the Dynkin diagram correspond to important features of the associated Lie algebra.

In mathematics, **G _{2}** is three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups. G

In mathematics, **F _{4}** is a Lie group and also its Lie algebra

In mathematics, **E _{6}** is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras , all of which have dimension 78; the same notation E

In mathematics, a **linear algebraic group** is a subgroup of the group of invertible matrices that is defined by polynomial equations. An example is the orthogonal group, defined by the relation where is the transpose of .

In mathematics, **E _{8}** is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E

In mathematics, **E _{7}** is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras

In mathematics, a **compact** (**topological**) **group** is a topological group whose topology realizes it as a compact topological space. Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory.

In mathematics, a **reductive group** is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group *G* over a perfect field is reductive if it has a representation that has a finite kernel and is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group *GL*(*n*) of invertible matrices, the special orthogonal group *SO*(*n*), and the symplectic group *Sp*(2*n*). **Simple algebraic groups** and (more generally) **semisimple algebraic groups** are reductive.

In mathematics, a Lie algebra is **semisimple** if it is a direct sum of simple Lie algebras..

In mathematics, a **symmetric space** is a Riemannian manifold whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis.

In mathematics, a **Hermitian symmetric space** is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds.

In mathematics, a **prehomogeneous vector space (PVS)** is a finite-dimensional vector space *V* together with a subgroup *G* of the general linear group GL(*V*) such that *G* has an open dense orbit in *V*. The term **prehomogeneous vector space** was introduced by Mikio Sato in 1970. These spaces have many applications in geometry, number theory and analysis, as well as representation theory. The irreducible PVS were classified first by Vinberg in his 1960 thesis in the special case when G is simple and later by Sato and Tatsuo Kimura in 1977 in the general case by means of a transformation known as "castling". They are subdivided into two types, according to whether the semisimple part of *G* acts prehomogeneously or not. If it doesn't then there is a homogeneous polynomial on *V* which is invariant under the semisimple part of *G*.

In the mathematical study of Lie algebras and Lie groups, a **Satake diagram** is a generalization of a Dynkin diagram introduced by Satake whose configurations classify simple Lie algebras over the field of real numbers. The Satake diagrams associated to a Dynkin diagram classify real forms of the complex Lie algebra corresponding to the Dynkin diagram.

In mathematics, the notion of a **real form** relates objects defined over the field of real and complex numbers. A real Lie algebra *g*_{0} is called a real form of a complex Lie algebra *g* if *g* is the complexification of *g*_{0}:

In mathematics, **Borel–de Siebenthal theory** describes the closed connected subgroups of a compact Lie group that have *maximal rank*, i.e. contain a maximal torus. It is named after the Swiss mathematicians Armand Borel and Jean de Siebenthal who developed the theory in 1949. Each such subgroup is the identity component of the centralizer of its center. They can be described recursively in terms of the associated root system of the group. The subgroups for which the corresponding homogeneous space has an invariant complex structure correspond to parabolic subgroups in the complexification of the compact Lie group, a reductive algebraic group.

In algebra, a **simple Lie algebra** is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan.

- 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.

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*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

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