View source for Simple Lie group

{{Short description|Connected non-abelian Lie group lacking nontrivial connected normal subgroups}}
{{About|the Killing-Cartan classification|a smaller list of groups that commonly occur in [[theoretical physics]]|Table of Lie groups|groups of dimension at most 3|Bianchi classification}}
{{more citations needed|date=April 2010}}
{{Lie groups|Simple}}
In mathematics, a '''simple Lie group''' is a [[connected space|connected]] [[nonabelian group|non-abelian]] [[Lie group]] ''G'' which does not have nontrivial connected [[normal subgroup]]s. The list of simple Lie groups can be used to read off the list of [[simple Lie algebra]]s and [[Riemannian symmetric space]]s.

Together with the commutative Lie group of the real numbers, <math>\mathbb{R}</math>, and that of the unit-magnitude complex numbers, [[Circle group|U(1)]] (the unit circle), simple Lie groups give the atomic "building 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'', <math>\mathbb{R}</math>) of ''n'' by ''n'' matrices with determinant equal to 1 is simple for all odd ''n''&nbsp;> 1, when it is isomorphic to the [[projective special linear group]].

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.

==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 group|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 <math>\mathbb{R}</math> 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 <math>\mathbb{R}</math> 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 [[covering space#universal covers|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 Lie algebra|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 group|simple as an abstract group]].

Simple Lie groups include many [[classical Lie group]]s, 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 [[list of simple Lie groups|classification]] of simple Lie groups that there exist also several [[exceptional object|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]].
<!-- ==Simple complex Lie groups== 
All (locally compact, connected) Lie groups are smooth [[manifolds]]. Mathematicians often study [[complex Lie group]]s, which are Lie groups with a [[Complex manifold|complex structure]] on the underlying manifold, which is required to be compatible with the group operations. A [[complex Lie group]] is called simple if it is connected as a topological space and its Lie algebra is simple as a [[complex Lie algebra]]. Note that the underlying Lie group may not be simple, although it will still be semisimple (see below). -->

As a counterexample, the [[general linear group]] is neither simple, nor [[semisimple Lie group|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 group]]s in even dimension. These have the matrix <math>-I</math> in the [[center (group theory)|center]], and this element is path-connected to the identity element, and so these groups evade the definition. Both of these are [[reductive group]]s.

==Related ideas==
===Semisimple Lie groups===
A '''semisimple''' Lie group is a connected Lie group so that its only [[closed subgroup|closed]] [[Connected space|connected]] [[Abelian group|abelian]] [[normal subgroup|normal]] subgroup is the trivial subgroup. Every simple Lie group is semisimple. More generally, any product of simple Lie groups is semisimple, and any quotient of a semisimple Lie group by a closed subgroup is semisimple. Every semisimple Lie group can be formed by taking a product of simple Lie groups and quotienting by a subgroup of its center. In other words, every semisimple Lie group is a [[central product]] of simple Lie groups. The semisimple Lie groups are exactly the Lie groups whose Lie algebras are [[semisimple Lie algebra]]s.

===Simple Lie algebras===
{{main|simple Lie algebra}}
The [[Lie group–Lie algebra correspondence|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 [[Triviality (mathematics)|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 diagram]]s, 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 form]]s 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===
{{Main|Symmetric space#Classification of Riemannian 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&nbsp;III, B&nbsp;I and D&nbsp;I for {{nowrap|1=''p'' = 2}}, D&nbsp;III, and C&nbsp;I, and the two exceptional ones are types E&nbsp;III and E&nbsp;VII of complex dimensions 16 and 27.

===Notation===
<math> \mathbb {R, C, H, O} </math>&nbsp; stand for the real numbers, complex numbers, [[quaternions]], and [[octonion]]s.

In the symbols such as ''E''<sub>6</sub><sup>&minus;26</sup> for the exceptional groups, the exponent &minus;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:
* [[Semisimple Lie algebra#Classification|Classification of simple complex Lie algebras]] The classification of simple Lie algebras over the complex numbers by [[Dynkin diagrams]].
* [[Satake diagram|Classification of simple real Lie algebras]] Each simple complex Lie algebra has several [[Real form (Lie theory)|real forms]], classified by additional decorations of its Dynkin diagram called [[Satake diagram]]s, after [[Ichirô Satake]].
* '''Classification of centerless simple Lie groups''' For every (real or complex) simple Lie algebra <math>\mathfrak{g}</math>, there is a unique "centerless" simple Lie group <math>G</math> whose Lie algebra is <math>\mathfrak{g}</math> and which has trivial [[Center (group theory)|center]].
* [[List of simple Lie groups|Classification of simple Lie groups]]
One can show that the [[fundamental group]] of any Lie group is a discrete [[Abelian group|commutative group]]. Given a (nontrivial) subgroup <math>K\subset \pi_1(G)</math> of the fundamental group of some Lie group <math>G</math>, one can use the theory of [[covering space]]s to construct a new group <math>\tilde{G}^K</math> with <math>K</math> 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 <math>K\subset \pi_1(G)</math> the full fundamental group, the resulting Lie group <math>\tilde{G}^{K = \pi_1(G)}</math> is the universal cover of the centerless Lie group <math>G</math>, and is simply connected. In particular, every (real or complex) Lie algebra also corresponds to a unique connected and [[Simply connected space|simply connected]] Lie group <math>\tilde{G}</math> with that Lie algebra, called the "simply connected Lie group" associated to <math>\mathfrak{g}.</math>

===Compact Lie groups===
{{Main|root system}}
Every simple complex Lie algebra has a unique real form whose corresponding centerless Lie group is [[Compact space|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]]).

[[File:Finite_Dynkin_diagrams.svg|Dynkin diagrams|480px]]

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==
A<sub>''r''</sub> has as its associated simply connected compact group the [[special unitary group]], [[Special unitary group|SU(''r'' + 1)]] and as its associated centerless compact group the projective unitary group [[Projective unitary group|PU(''r'' + 1)]].

B<sub>''r''</sub> has as its associated centerless compact groups the odd [[special orthogonal group]]s, [[Special orthogonal group|SO(2''r'' + 1)]]. This group is not simply connected however: its universal (double) cover is the [[spin group]].

C<sub>''r''</sub> has as its associated simply connected group the group of [[symplectic group|unitary symplectic matrices]], [[Symplectic group|Sp(''r'')]] and as its associated centerless group the Lie group {{math|1=PSp(''r'') = Sp(''r'')/{{mset|I, −I}}}} of projective unitary symplectic matrices. The symplectic groups have a double-cover by the [[metaplectic group]].

D<sub>''r''</sub> has as its associated compact group the even [[special orthogonal group]]s, [[Special orthogonal group|SO(2''r'')]] and as its associated centerless compact group the projective special orthogonal group {{math|1=PSO(2''r'') = SO(2''r'')/{{mset|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<sub>2</sub> is two isolated nodes, the same as A<sub>1</sub> &cup; A<sub>1</sub>, 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#Pairs of unit quaternions as rotations in 4D space|quaternions and spatial rotation]]. Thus SO(4) is not a simple group. Also, the diagram D<sub>3</sub> is the same as A<sub>3</sub>, corresponding to a covering map homomorphism from SU(4) to SO(6).

In addition to the four families ''A''<sub>''i''</sub>, ''B''<sub>''i''</sub>, ''C''<sub>''i''</sub>, and ''D''<sub>''i''</sub> above, there are five so-called exceptional Dynkin diagrams [[G2 (mathematics)|G<sub>2</sub>]], [[F4 (mathematics)|F<sub>4</sub>]], [[E6 (mathematics)|E<sub>6</sub>]], [[E7 (mathematics)|E<sub>7</sub>]], and [[E8 (mathematics)|E<sub>8</sub>]]; 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 object]]s. For example, the group associated to G<sub>2</sub> is the automorphism group of the [[octonion]]s, and the group associated to F<sub>4</sub> is the automorphism group of a certain [[Albert algebra]].

See also [[E7 1/2|{{tmath|\color{Blue} E_{7\frac 1 2} }}]].

== List ==

=== Abelian ===

{{See also|Abelian group}}

{| class="wikitable"
|-
! width=100|
! Dimension
! Outer automorphism group
! Dimension of symmetric space
! Symmetric space
! Remarks
|-class="sorttop"
! <math>\mathbb{R}</math> (Abelian)
| 1
| <math>\mathbb{R}^*</math>
| 1
| <math>\mathbb{R}</math>
| {{Ref|Note†|†}}
|}

====Notes====
:{{Note|Note†|†}} The group <math>\mathbb{R}</math> 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 <math>\mathbb{R}</math> is the only such non-compact symmetric space without a compact dual (although it has a compact quotient ''S''<sup>1</sup>).

=== Compact ===

{{See also|Compact group}}

{| class="wikitable sortable"
|-
! width=100|
! Dimension
! Real rank
! Fundamental<br>group
! class="unsortable" | Outer automorphism<br>group
! class="unsortable" | Other names
! class="unsortable" | Remarks
|-
! ''A''<sub>''n''</sub> ({{math|''n'' ≥ 1}}) compact
| ''n''(''n'' + 2)
| 0
| Cyclic,<br/>order {{math|''n'' + 1}}
| 1 if {{math|1=''n'' = 1}},<br/>2 if {{math|''n'' > 1}}.
| '''[[projective special unitary group]]'''<br>{{math|PSU(''n'' + 1)}}
| ''A''<sub>1</sub> is the same as ''B''<sub>1</sub> and ''C''<sub>1</sub>
|-
! ''B''<sub>''n''</sub> ({{math|''n'' ≥ 2}}) compact
| ''n''(2''n'' + 1)
| 0
| 2
| 1
| '''[[special orthogonal group]]'''<br>SO<sub>2''n''+1</sub>(''R'')
| ''B''<sub>1</sub> is the same as ''A''<sub>1</sub> and ''C''<sub>1</sub>.<br>''B''<sub>2</sub> is the same as ''C''<sub>2</sub>.
|-
! ''C''<sub>''n''</sub> ({{math|''n'' ≥ 3}}) compact
| ''n''(2''n'' + 1)
| 0
| 2
| 1
| '''projective [[compact symplectic group]]'''<br>PSp(''n''), PSp(2''n''), PUSp(''n''), PUSp(2''n'')
| Hermitian. Complex structures of ''H''<sup>''n''</sup>. Copies of complex projective space in quaternionic projective space.
|-
! ''D''<sub>''n''</sub> ({{math|''n'' ≥ 4}}) compact
| ''n''(2''n'' &minus; 1)
| 0
| Order 4 (cyclic when ''n'' is odd).
| 2 if {{math|''n'' > 4}},<br/>''S''<sub>3</sub> if {{math|1=''n'' = 4}}
|style="white-space:math"| '''projective special [[orthogonal group]]'''<br>PSO<sub>2''n''</sub>(''R'')
| ''D''<sub>3</sub> is the same as ''A''<sub>3</sub>, ''D''<sub>2</sub> is the same as ''A''<sub>1</sub><sup>2</sup>, and ''D''<sub>1</sub> is abelian.
|-
! ''E''<sub>6</sub><sup>&minus;78</sup> compact
| 78
| 0
| 3
| 2
| 
| 
|-
! ''E''<sub>7</sub><sup>&minus;133</sup> compact
| 133
| 0
| 2
| 1
| 
| 
|-
! ''E''<sub>8</sub><sup>&minus;248</sup> compact
| 248
| 0
| 1
| 1
| 
| 
|-
! ''F''<sub>4</sub><sup>&minus;52</sup> compact
| 52
| 0
| 1
| 1
| 
| 
|-
! ''G''<sub>2</sub><sup>&minus;14</sup> compact
| 14
| 0
| 1
| 1
| 
| This is the automorphism group of the Cayley algebra.
|}

=== Split ===

{{See also|Split Lie algebra}}
{{sort-under}}
{| class="wikitable sortable sort-under"
|-
! 
! Dimension
! {{verth|va=middle|Real rank}}
! Maximal compact<br>subgroup
! Fundamental<br>group
! class="unsortable" | Outer auto&shy;morphism<br>group
! class="unsortable" | Other names
! {{verth|va=middle|Dimension of<br>symmetric space}}
! class="unsortable" | Compact<br>symmetric space
! class="unsortable" | Non-Compact<br>symmetric space
! class="unsortable" | Remarks
|-
! {{verth|va=middle|''A''<sub>''n''</sub> I (''n'' ≥ 1) split}}
| ''n''(''n'' + 2)
| ''n''
| ''D''<sub>''n''/2</sub> or ''B''<sub>(''n''&minus;1)/2</sub>
| Infinite cyclic if ''n'' = 1<br>2 if ''n'' ≥ 2
| 1 if ''n'' = 1<br>2 if ''n'' ≥ 2.
| '''[[projective special linear group]]'''<br>PSL<sub>''n''+1</sub>(R)
| {{math|{{sfrac|''n''(''n'' + 3)|2}}}}
| Real structures on ''C''<sup>''n''+1</sup> or set of RP<sup>''n''</sup> in CP<sup>''n''</sup>. Hermitian if {{math|1=''n'' = 1}}, in which case it is the 2-sphere. 
| Euclidean structures on ''R''<sup>''n''+1</sup>. Hermitian if {{math|1=''n'' = 1}}, when it is the upper half plane or unit complex disc.
| 
|-
! {{verth|va=middle|''B''<sub>''n''</sub> I (''n'' ≥ 2) split}}
| ''n''(2''n'' + 1)
| ''n''
| SO(''n'')SO(''n''+1)
| Non-cyclic, order 4
| 1
| identity component of '''[[indefinite orthogonal group|special orthogonal group]]'''<br>SO(''n'',''n''+1)
| {{math|''n''(''n'' + 1)}}
|
|
| ''B''<sub>1</sub> is the same as ''A''<sub>1</sub>.
|-
! {{verth|va=middle|''C''<sub>''n''</sub> I (''n'' ≥ 3) split}}
| ''n''(2''n'' + 1)
| ''n''
| ''A''<sub>''n''&minus;1</sub>''S''<sup>1</sup>
| Infinite cyclic
| 1
| '''projective [[symplectic group]]'''<br>PSp<sub>2''n''</sub>(''R''), PSp(2''n'',''R''), PSp(2''n''), PSp(''n'',''R''), PSp(''n'')
| {{math|''n''(''n'' + 1)}}
| Hermitian. Complex structures of ''H''<sup>''n''</sup>. Copies of complex projective space in quaternionic projective space. 
| Hermitian. Complex structures on ''R''<sup>2''n''</sup> compatible with a symplectic form. Set of complex hyperbolic spaces in quaternionic hyperbolic space. Siegel upper half space.
| ''C''<sub>2</sub> is the same as ''B''<sub>2</sub>, and ''C''<sub>1</sub> is the same as ''B''<sub>1</sub> and ''A''<sub>1</sub>.
|-
! {{verth|va=middle|''D''<sub>''n''</sub> I (''n'' ≥ 4) split}}
| ''n''(2''n'' - 1)
| ''n''
| SO(''n'')SO(''n'')
| Order 4 if ''n'' odd,<br/>8 if ''n'' even
| 2 if {{math|''n'' > 4}},<br/>''S''<sub>3</sub> if {{math|1=''n'' = 4}}
| identity component of '''projective [[indefinite orthogonal group|special orthogonal group]]'''<br>PSO(''n'',''n'')
| ''n''<sup>2</sup>
|
|
| ''D''<sub>3</sub> is the same as ''A''<sub>3</sub>, ''D''<sub>2</sub> is the same as ''A''<sub>1</sub><sup>2</sup>, and ''D''<sub>1</sub> is abelian.
|-
! {{verth|va=middle|''E''<sub>6</sub><sup>6</sup> I split}}
| 78
| 6
| ''C''<sub>4</sub>
| Order 2
| Order 2
| E I
| 42 
| 
| 
| 
|-
! {{verth|va=middle|''E''<sub>7</sub><sup>7</sup> V split}}
| 133
| 7
| ''A''<sub>7</sub>
| Cyclic, order 4
| Order 2
| 
| 70
| 
| 
| 
|-
! {{verth|va=middle|''E''<sub>8</sub><sup>8</sup> VIII split}}
| 248
| 8
| ''D''<sub>8</sub>
| 2
| 1
| E VIII
| 128
| 
| 
| @ [[E8 (mathematics)|E8]]
|-
! {{verth|va=middle|''F''<sub>4</sub><sup>4</sup> I split}}
| 52
| 4
| ''C''<sub>3</sub> × ''A''<sub>1</sub>
| Order 2
| 1
| F I
| 28
| Quaternionic projective planes in Cayley projective plane.
| Hyperbolic quaternionic projective planes in hyperbolic Cayley projective plane.
| 
|-
! {{verth|va=middle|''G''<sub>2</sub><sup>2</sup> I split}}
| 14
| 2
| ''A''<sub>1</sub> × ''A''<sub>1</sub>
| 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 ===

{{See also|Complex Lie group}}
{{sort-under}}
{| class="wikitable sortable sort-under"
|-
!
! Real dimension
! {{verth|Real rank}}
! {{verth|Maximal compact<br>subgroup}}
! Fundamental<br>group
! class="unsortable" | Outer auto&shy;morphism<br>group
! class="unsortable" | Other names
! Dimension of<br>symmetric space
! class="unsortable" | Compact<br>symmetric space
! class="unsortable" | Non-Compact<br>symmetric space
|-
! ''A''<sub>''n''</sub><br/>(''n'' ≥ 1) complex
| 2''n''(''n'' + 2)
| ''n''
| ''A''<sub>''n''</sub>
| Cyclic, order {{math|''n'' + 1}}
| 2 if {{math|1=''n'' = 1}},<br/>4 (noncyclic) if {{math|''n'' ≥ 2}}.
| '''[[projective special linear group|projective complex special linear group]]'''<br>PSL<sub>''n''+1</sub>(''C'')
| ''n''(''n'' + 2)
| Compact group ''A''<sub>''n''</sub>
| Hermitian forms on ''C''<sup>''n''+1</sup>
with fixed volume.
|-
! ''B''<sub>n</sub><br/>(''n'' ≥ 2) complex
| 2''n''(2''n'' + 1)
| ''n''
| ''B''<sub>''n''</sub>
| 2
| Order 2 (complex conjugation)
| '''complex [[special orthogonal group]]'''<br>SO<sub>2''n''+1</sub>('''C''')
| ''n''(2''n'' + 1)
| Compact group ''B''<sub>n</sub>
| 
|-
! ''C''<sub>''n''</sub><br/>(''n'' ≥ 3) complex
| 2''n''(2''n'' + 1)
| ''n''
| ''C''<sub>''n''</sub>
| 2
| Order 2 (complex conjugation)
| '''projective complex [[symplectic group]]'''<br>PSp<sub>2''n''</sub>('''C''')
| ''n''(2''n'' + 1)
| Compact group ''C''<sub>n</sub>
| 
|-
! ''D''<sub>''n''</sub><br/>(''n'' ≥ 4) complex
| 2''n''(2''n'' &minus; 1)
| ''n''
| ''D''<sub>''n''</sub>
| Order 4 (cyclic when ''n'' is odd)
| Noncyclic of order 4 for {{math|''n'' > 4}}, or the product of a group of order 2 and the symmetric group ''S''<sub>3</sub> when {{math|1=''n'' = 4}}.
| '''projective complex special orthogonal group'''<br>PSO<sub>2''n''</sub>('''C''')
| ''n''(2''n'' &minus; 1)
| Compact group ''D''<sub>n</sub>
| 
|-
! ''E''<sub>6</sub> complex
| 156
| 6
| ''E''<sub>6</sub>
| 3
| Order 4 (non-cyclic)
| 
| 78
| Compact group ''E''<sub>6</sub>
| 
|-
! ''E''<sub>7</sub> complex
| 266
| 7
| ''E''<sub>7</sub>
| 2
| Order 2 (complex conjugation)
| 
| 133
| Compact group ''E''<sub>7</sub>
| 
|-
! ''E''<sub>8</sub> complex
| 496
| 8
| ''E''<sub>8</sub>
| 1
| Order 2 (complex conjugation)
| 
| 248
| Compact group ''E''<sub>8</sub>
| 
|-
! ''F''<sub>4</sub> complex
| 104
| 4
| ''F''<sub>4</sub>
| 1
| 2
| 
| 52
| Compact group ''F''<sub>4</sub>
| 
|-
! ''G''<sub>2</sub> complex
| 28
| 2
| ''G''<sub>2</sub> 
| 1
| Order 2 (complex conjugation)
| 
| 14
| Compact group ''G''<sub>2</sub>
| 
|}

=== Others ===
{{sort-under}}
{| class="wikitable sortable sort-under"
|-
! 
! Dimension
! Real rank
! Maximal compact<br>subgroup
! Fundamental<br>group
! class="unsortable" | Outer automorphism<br>group
! class="unsortable" | Other names
! Dimension of<br>symmetric space
! class="unsortable" | Compact<br>symmetric space
! class="unsortable" | Non-Compact<br>symmetric space
! class="unsortable" | Remarks
|-
! ''A''<sub>2''n''&minus;1</sub> II<br>(''n'' ≥ 2)
| {{math|(2''n'' &minus; 1)(2''n'' + 1)}}
| ''n'' &minus; 1
| ''C''<sub>''n''</sub>
| Order 2
| 
| SL<sub>''n''</sub>(''H''), SU<sup>∗</sup>(2''n'')
| {{hs|zzzzzz <!-- when sorted, comes after numbers -->}}{{math|(''n'' &minus; 1)(2''n'' + 1)}}
| Quaternionic structures on ''C''<sup>2''n''</sup> compatible with the Hermitian structure
| Copies of [[quaternionic hyperbolic space]] (of dimension {{math|''n'' &minus; 1}}) in [[complex hyperbolic space]] (of dimension {{math|2''n'' &minus; 1}}).
| 
|-
!style="white-space:nowrap"| ''A''<sub>''n''</sub> III<br>(''n'' ≥ 1)<br>''p'' + ''q'' = ''n'' + 1<br>(1 ≤ ''p'' ≤ ''q'')
| ''n''(''n'' + 2)
| ''p''
| ''A''<sub>''p''&minus;1</sub>''A''<sub>''q''&minus;1</sub>''S''<sup>1</sup>
| 
| 
| SU(''p'',''q''), A III
| 2''pq''
|style="white-space:nowrap"| [[Hermitian symmetric space|Hermitian]].<br>Grassmannian of ''p'' subspaces of ''C''<sup>''p''+''q''</sup>.<br>If ''p'' or ''q'' is 2; [[Quaternion-Kähler symmetric space|quaternion-Kähler]]
|style="white-space:nowrap"| Hermitian.<br>Grassmannian of maximal positive definite<br>subspaces of ''C''<sup>''p'',''q''</sup>.<br>If ''p'' or ''q'' is 2, quaternion-Kähler
|style="white-space:nowrap"|  If ''p''=''q''=1, split<br>If {{abs|''p''−''q''}} ≤ 1, [[quasi-split]]
|-
! ''B''<sub>''n''</sub> I<br>(''n'' > 1)<br>''p''+''q'' = 2''n''+1
| ''n''(2''n'' + 1)
| min(''p'',''q'')
| SO(''p'')SO(''q'')
| 
| 
| [[indefinite orthogonal group|SO(''p'',''q'')]]
| ''pq''
|style="white-space:nowrap"| Grassmannian of ''R''<sup>''p''</sup>s in ''R''<sup>''p''+''q''</sup>.<br>If ''p'' or ''q'' is 1, Projective space<br>If ''p'' or ''q'' is 2; Hermitian<br>If ''p'' or ''q'' is 4, quaternion-Kähler
|style="white-space:nowrap"| Grassmannian of positive definite ''R''<sup>''p''</sup>s in ''R''<sup>''p'',''q''</sup>.<br>If ''p'' or ''q'' is 1,  Hyperbolic space<br>If ''p'' or ''q'' is 2, Hermitian<br>If ''p'' or ''q'' is 4, quaternion-Kähler
|style="white-space:nowrap"| If {{abs|''p''−''q''}} ≤ 1, split.
|-
! ''C''<sub>''n''</sub> II<br>(''n'' > 2)<br>''n'' = ''p''+''q''<br>(1 ≤ ''p'' ≤ ''q'')
| ''n''(2''n'' + 1)
| min(''p'',''q'')
| ''C''<sub>''p''</sub>''C''<sub>''q''</sub>
| Order 2
|style="white-space:nowrap"| 1 if ''p'' ≠ ''q'', 2 if ''p'' = ''q''.
| Sp<sub>2''p'',2''q''</sub>(R)
|  4''pq''
|style="white-space:nowrap"| Grassmannian of ''H''<sup>''p''</sup>s in ''H''<sup>''p''+''q''</sup>.<br>If ''p'' or ''q'' is 1, quaternionic projective space<br>in which case it is quaternion-Kähler.
|style="white-space:nowrap"| ''H''<sup>''p''</sup>s in ''H''<sup>''p'',''q''</sup>.<br>If ''p'' or ''q'' is 1, quaternionic hyperbolic space<br>in which case it is quaternion-Kähler.
| 
|-
! ''D''<sub>''n''</sub> I<br>(''n'' ≥ 4)<br>''p''+''q'' = 2''n''
| ''n''(2''n'' &minus; 1)
| min(''p'',''q'')
| SO(''p'')SO(''q'')
| 
| If ''p'' and ''q'' ≥ 3, order 8.
| SO(''p'',''q'')
| ''pq''
|style="white-space:nowrap"| Grassmannian of ''R''<sup>''p''</sup>s in ''R''<sup>''p''+''q''</sup>.<br>If ''p'' or ''q'' is 1, Projective space<br>If ''p'' or ''q'' is 2 ; Hermitian<br>If ''p'' or ''q'' is 4, quaternion-Kähler
|style="white-space:nowrap"| Grassmannian of positive definite ''R''<sup>''p''</sup>s in ''R''<sup>''p'',''q''</sup>.<br>If ''p'' or ''q'' is 1, Hyperbolic Space<br>If ''p'' or ''q'' is 2, Hermitian<br>If ''p'' or ''q'' is 4, quaternion-Kähler
|style="white-space:nowrap"| If {{math|1=''p'' = ''q''}}, split<br>If {{abs|''p''−''q''}} ≤ 2, quasi-split
|-
! ''D''<sub>''n''</sub> III<br>(''n'' ≥ 4)
| ''n''(2''n'' &minus; 1)
| ⌊''n''/2⌋
| ''A''<sub>''n''&minus;1</sub>''R''<sup>1</sup>
| Infinite cyclic
| Order 2
| ''SO''<sup>*</sup>(2n)
| ''n''(''n'' &minus; 1)
| Hermitian.<br>Complex structures on R<sup>2''n''</sup> compatible with the Euclidean structure.
| Hermitian.<br>Quaternionic quadratic forms on R<sup>2''n''</sup>.
| 
|-
! ''E''<sub>6</sub><sup>2</sup> II<br>(quasi-split)
| 78
| 4
| ''A''<sub>5</sub>''A''<sub>1</sub>
| Cyclic, order 6
| Order 2
| E II
| 40
| Quaternion-Kähler.
| Quaternion-Kähler.
| Quasi-split but not split.
|-
! ''E''<sub>6</sub><sup>&minus;14</sup> III
| 78
| 2
| ''D''<sub>5</sub>''S''<sup>1</sup>
| Infinite cyclic
| Trivial
| E III
| 32
| Hermitian.<br>Rosenfeld elliptic projective plane over the complexified Cayley numbers.
| Hermitian.<br>Rosenfeld hyperbolic projective plane over the complexified Cayley numbers.
| 
|-
! ''E''<sub>6</sub><sup>&minus;26</sup> IV
| 78
| 2
| ''F''<sub>4</sub>
| Trivial
| Order 2
| E IV
| 26
| Set of [[Cayley projective plane]]s in the projective plane over the complexified Cayley numbers. 
| Set of Cayley hyperbolic planes in the hyperbolic plane over the complexified Cayley numbers.
| 
|-
! ''E''<sub>7</sub><sup>&minus;5</sup> VI
| 133
| 4
| ''D''<sub>6</sub>''A''<sub>1</sub>
| Non-cyclic, order 4
| Trivial
| ''E'' VI
| 64
| Quaternion-Kähler.
| Quaternion-Kähler.
| 
|-
! ''E''<sub>7</sub><sup>&minus;25</sup> VII
| 133
| 3
| ''E''<sub>6</sub>''S''<sup>1</sup>
| Infinite cyclic
| Order 2
| E VII
| 54
| Hermitian.
| Hermitian.
|
|-
! ''E''<sub>8</sub><sup>&minus;24</sup> IX
| 248
| 4
| ''E''<sub>7</sub> × ''A''<sub>1</sub>
| Order 2
| 1
| E IX
| 112
| Quaternion-Kähler.
| Quaternion-Kähler.
| 
|-
! ''F''<sub>4</sub><sup>&minus;20</sup> II
| 52
| 1
| ''B''<sub>4</sub> (Spin<sub>9</sub>('''R'''))
| Order 2
| 1
| F II
| 16
| Cayley 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.

{| class="wikitable sortable"
|-
!Dim
!Groups
!
!Symmetric space
!Compact dual
!Rank
!Dim
|-
|1
|{{mathbb|R}}, ''S''<sup>1</sup> = U(1) = SO<sub>2</sub>({{mathbb|R}}) = Spin(2)
|Abelian
|Real line
|
|0
|1
|-
|3
| ''S''<sup>3</sup> = Sp(1) = SU(2)=Spin(3), SO<sub>3</sub>({{mathbb|R}}) = PSU(2) 
|Compact
|
|
|
|
|-
|3
|SL<sub>2</sub>({{mathbb|R}}) = Sp<sub>2</sub>({{mathbb|R}}), SO<sub>2,1</sub>({{mathbb|R}})
|Split, Hermitian, hyperbolic
|Hyperbolic plane <math>\mathbb{H}^2</math>
|Sphere ''S''<sup>2</sup>
|1
|2
|-
|6
|SL<sub>2</sub>({{mathbb|C}}) = Sp<sub>2</sub>({{mathbb|C}}), SO<sub>3,1</sub>({{mathbb|R}}), SO<sub>3</sub>({{mathbb|C}})
|Complex
|Hyperbolic space <math>\mathbb{H}^3</math>
|Sphere ''S''<sup>3</sup>
|1
|3
|-
|8
|SL<sub>3</sub>({{mathbb|R}})
|Split
|Euclidean structures on <math>\mathbb{R}^3</math>
|Real structures on <math>\mathbb{C}^3</math>
|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<sub>5</sub>({{mathbb|R}})
|Compact
|
|
|
|
|-
|10
|SO<sub>4,1</sub>({{mathbb|R}}), Sp<sub>2,2</sub>({{mathbb|R}})
|Hyperbolic, quaternionic
|Hyperbolic space <math>\mathbb{H}^4</math>
|Sphere ''S''<sup>4</sup>
|1
|4
|-
|10
|SO<sub>3,2</sub>({{mathbb|R}}), Sp<sub>4</sub>({{mathbb|R}})
|Split, Hermitian
|Siegel upper half space
|Complex structures on <math>\mathbb{H}^2</math>
|2
|6
|-
|14
|''G''<sub>2</sub> 
|Compact
|
|
|
|
|-
|14
|''G''<sub>2</sub> 
|Split, quaternionic
|Non-division quaternionic subalgebras of non-division octonions
|Quaternionic subalgebras of octonions
|2
|8
|-
|15
|SU(4) = Spin(6), SO<sub>6</sub>({{mathbb|R}})
|Compact
|
|
|-
|15
|SL<sub>4</sub>({{mathbb|R}}), SO<sub>3,3</sub>({{mathbb|R}})
|Split
|{{mathbb|R}}<sup>3</sup> in {{mathbb|R}}<sup>3,3</sup>
|Grassmannian ''G''(3,3)
|3
|9
|-
|15
|SU(3,1)
|Hermitian
|Complex hyperbolic space
|Complex projective space
|1
|6
|-
|15
|SU(2,2), SO<sub>4,2</sub>({{mathbb|R}})
|Hermitian, quasi-split, quaternionic
|{{mathbb|R}}<sup>2</sup> in {{mathbb|R}}<sup>2,4</sup>
|Grassmannian ''G''(2,4)
|2
|8
|-
|15
|SL<sub>2</sub>({{mathbb|H}}), SO<sub>5,1</sub>({{mathbb|R}})
|Hyperbolic
|Hyperbolic space <math>\mathbb{H}^5</math>
|Sphere ''S''<sup>5</sup>
|1
|5
|-
|16
|SL<sub>3</sub>({{mathbb|C}})
|Complex
|
|SU(3)
|2
|8
|-
|20
|SO<sub>5</sub>({{mathbb|C}}), Sp<sub>4</sub>({{mathbb|C}})
|Complex
|
|Spin<sub>5</sub>({{mathbb|R}})
|2
|10
|-
|21
|SO<sub>7</sub>({{mathbb|R}})
|Compact
|
|
|-
|21
|SO<sub>6,1</sub>({{mathbb|R}})
|Hyperbolic
|Hyperbolic space <math>\mathbb{H}^6</math>
|Sphere ''S''<sup>6</sup>
|-
|21
|SO<sub>5,2</sub>({{mathbb|R}})
|Hermitian
|
|
|-
|21
|SO<sub>4,3</sub>({{mathbb|R}})
|Split, quaternionic
|
|
|-
|21
|Sp(3)
|Compact
|
|
|-
|21
|Sp<sub>6</sub>({{mathbb|R}})
|Split, hermitian
|
|
|-
|21
|Sp<sub>4,2</sub>({{mathbb|R}})
|Quaternionic
|
|
|-
|24
|SU(5)
|Compact
|
|
|-
|24
|SL<sub>5</sub>({{mathbb|R}})
|Split
|
|
|-
|24
|SU<sub>4,1</sub>
|Hermitian
|
|
|-
|24
|SU<sub>3,2</sub>
|Hermitian, quaternionic
|
|
|-
|28
|SO<sub>8</sub>({{mathbb|R}})
|Compact
|
|
|-
|28
|SO<sub>7,1</sub>({{mathbb|R}})
|Hyperbolic
|Hyperbolic space <math>\mathbb{H}^7</math>
|Sphere ''S''<sup>7</sup>
|-
|28
|SO<sub>6,2</sub>({{mathbb|R}})
|Hermitian
|
|
|-
|28
|SO<sub>5,3</sub>({{mathbb|R}})
|Quasi-split
|
|
|-
|28
|SO<sub>4,4</sub>({{mathbb|R}})
|Split, quaternionic
|
|
|-
|28
|SO<sup>∗</sup><sub>8</sub>({{mathbb|R}})
|Hermitian
|
|
|-
|28
|''G''<sub>2</sub>({{mathbb|C}})
|Complex
|
|
|-
|30
|SL<sub>4</sub>({{mathbb|C}})
|Complex
|
|
<!-- to do: 24, 28, 30, 35, 36, 42, 45, 48, 52, 55, 56, 63, 66, 70, 72, 78, 80, 90, 91, 96, 99, 104-->
|}

==Simply laced groups<!--'Simple Lie groups' redirects here-->==
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==
*[[Cartan matrix]]
*[[Coxeter matrix]]
*[[Weyl group]]
*[[Coxeter group]]
*[[Kac–Moody algebra]]
*[[Catastrophe theory]]
*[[Table of Lie groups]]
*[[Classification of low-dimensional real Lie algebras]]

== References ==
{{Reflist}}
* {{Cite book
| publisher = CRC Press
| isbn = 0-8247-1326-5
| last1 = Jacobson
| first1 = Nathan
| author1-link = Nathan Jacobson
| title = Exceptional Lie Algebras
| date = 1971
}}
*{{cite book |author1-link=William Fulton (mathematician) |last1=Fulton |first1=William |author2-link=Joe Harris (mathematician) |last2=Harris |first2=Joe |year=2004   |title=Representation Theory: A First Course |publisher=Springer |doi=10.1007/978-1-4612-0979-9 |isbn=978-1-4612-0979-9}}

==Further reading==
* 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}}

{{Authority control}}

{{DEFAULTSORT:Simple Lie Group}}
[[Category:Lie groups]]
[[Category:Lie algebras]]