Editing Simple Lie group (section)

== 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.
|
|}