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{{Short description|Simple Lie group; the automorphism group of the octonions}}
{{DISPLAYTITLE:G<sub>2</sub> (mathematics)}}
{{Group theory sidebar |Topological}}
{{Lie groups |Simple}}

In [[mathematics]], '''G<sub>2</sub>''' is three simple [[Lie group]]s (a complex form, a compact real form and a split real form), their [[Lie algebra]]s <math>\mathfrak{g}_2,</math> as well as some [[algebraic group]]s. They are the smallest of the five exceptional [[simple Lie group]]s. G<sub>2</sub> has rank 2 and dimension 14. It has two [[fundamental representation]]s, with dimension 7 and 14.

The compact form of G<sub>2</sub> can be described as the [[automorphism group]] of the [[Octonion|octonion algebra]] or, equivalently, as the subgroup of SO(7) that preserves any chosen particular vector in its 8-dimensional [[Real representation|real]] [[spinor]] [[Group representation|representation]] (a [[spin representation]]).

== History ==

The Lie algebra <math>\mathfrak{g}_2</math>, being the smallest exceptional simple Lie algebra, was the first of these to be discovered in the attempt to classify simple Lie algebras.  On May 23, 1887, [[Wilhelm Killing]] wrote a letter to [[Friedrich Engel (mathematician)|Friedrich Engel]] saying that he had found a 14-dimensional simple Lie algebra, which we now call <math>\mathfrak{g}_2</math>.<ref>{{cite journal
 | last = Agricola | first = Ilka | author-link = Ilka Agricola
 | issue = 8
 | journal = Notices of the American Mathematical Society
 | mr = 2441524
 | pages = 922–929
 | title = Old and new on the exceptional group ''G''<sub>2</sub>
 | url = https://www.ams.org/notices/200808/tx080800922p.pdf
 | volume = 55
 | year = 2008}}</ref>

In 1893, [[Élie Cartan]] published a note describing an open set in <math>\mathbb{C}^5</math> equipped with a 2-dimensional [[distribution (differential geometry)|distribution]]—that is, a smoothly varying field of 2-dimensional subspaces of the tangent space—for which the Lie algebra <math>\mathfrak{g}_2</math> appears as the infinitesimal symmetries.<ref>{{cite journal|author=Élie Cartan|title=Sur la structure des groupes simples finis et continus|journal=C. R. Acad. Sci.|volume=116|year=1893|pages=784–786}}</ref> In the same year, in the same journal, Engel noticed the same thing.   Later it was discovered that the 2-dimensional distribution is closely related to a ball rolling on another ball.   The space of configurations of the rolling ball is 5-dimensional, with a 2-dimensional distribution that describes motions of the ball where it rolls without slipping or twisting.<ref>{{cite journal| title = G<sub>2</sub> and the "rolling distribution" | author = Gil Bor and Richard Montgomery |journal =L'Enseignement Mathématique|volume =55|year=2009|pages=157–196|doi=10.4171/lem/55-1-8|arxiv=math/0612469| s2cid = 119679882 }}</ref><ref>{{cite journal| title = G<sub>2</sub> and the rolling ball | author = John Baez and John Huerta |arxiv=1205.2447|journal =Trans. Amer. Math. Soc.|volume =366| issue = 10 |year=2014|pages=5257–5293|doi=10.1090/s0002-9947-2014-05977-1}}</ref>

In 1900, Engel discovered that a generic antisymmetric trilinear form (or 3-form) on a 7-dimensional complex vector space is preserved by a group isomorphic to the complex form of G<sub>2</sub>.<ref>{{cite journal|author=Friedrich Engel|title=Ein neues, dem linearen Komplexe analoges Gebilde|journal=Leipz. Ber.|volume=52|year=1900|pages=63–76,220–239}}</ref>

In 1908 Cartan mentioned that the automorphism group of the octonions is a 14-dimensional simple Lie group.<ref>{{cite book|author=Élie Cartan|chapter= Nombres complexes|title=Encyclopedie des Sciences Mathematiques|publisher=Gauthier-Villars|location=Paris|year= 1908|pages = 329–468}}</ref>  In 1914 he stated that this is the compact real form of G<sub>2</sub>.<ref>{{citation|author=Élie Cartan|title=Les groupes reels simples finis et continus|journal=Ann. Sci. École Norm. Sup.|volume=31|year=1914|pages=255–262}}</ref>

In older books and papers, G<sub>2</sub> is sometimes denoted by E<sub>2</sub>.

==Real forms==
There are 3 simple real Lie algebras associated with this root system:

*The underlying real Lie algebra of the complex Lie algebra G<sub>2</sub> has dimension 28. It has complex conjugation as an outer automorphism and is simply connected. The maximal compact subgroup of its associated group is the compact form of G<sub>2</sub>.
*The Lie algebra of the compact form is 14-dimensional. The associated Lie group has no outer automorphisms, no center, and is simply connected and compact.
*The Lie algebra of the  non-compact (split) form has dimension 14. The associated simple Lie group has fundamental group of order 2 and its [[outer automorphism group]] is the trivial group. Its maximal compact subgroup is {{nowrap|SU(2) × SU(2)/(−1,−1)}}. It has a non-algebraic double cover that is simply connected.

== Algebra ==

===Dynkin diagram and Cartan matrix ===
The [[Dynkin diagram]] for ''G''<sub>2</sub> is given by [[Image:Dynkin diagram G2.png|Dynkin diagram of G 2]].

Its [[Cartan matrix]] is:

: <math>
\left [\begin{array}{rr}
2 & -3 \\
-1 & 2
\end{array}\right]
</math>

=== Roots of G<sub>2</sub> ===
{| class=wikitable width=480 
|- valign=top
|[[File:Root system G2.svg|160px]]<BR>The 12 vector [[root system]] of G<sub>2</sub> in 2 dimensions.
|[[File:3-cube t1.svg|160px]]<BR>The A<sub>2</sub> [[Coxeter plane]] projection of the 12 vertices of the [[cuboctahedron]] contain the same 2D vector arrangement.
|[[Image:G2Coxeter.svg|160px]]<BR>Graph of G2 as a subgroup of F4 and E8 projected into the Coxeter plane
|}

A set of '''simple roots''' for {{Dynkin2|node_n1|6a|node_n2}} can be read directly from the Cartan matrix above. These are (2,&minus;3) and (&minus;1, 2), however the integer lattice spanned by those is not the one pictured above (from obvious reason: the hexagonal lattice on the plane cannot be generated by integer vectors). The diagram above is obtained from a different pair roots: <math>\alpha = \left( 1, 0 \right)</math> and <math display="inline">\beta = \sqrt{3}\left(\cos{\frac{5\pi}{6}},\sin{\frac{5\pi}{6}}\right) = \frac{1}{2}\left(-3,\sqrt{3} \right)</math>. 

The remaining [[Positive roots|(positive) roots]] are <math display="inline">A = \alpha + \beta,\, B = 3\alpha + \beta,\, \alpha + A = 2\alpha + \beta \,\,{\rm and }\,\, \beta + B = 3\alpha + 2\beta</math>. 

Although they do [[Linear span|span]] a 2-dimensional space, as drawn, it is much more symmetric to consider them as [[Vector space|vectors]] in a 2-dimensional subspace of a three-dimensional space. In this identification α corresponds to e₁&minus;e₂, β to &minus;e₁ + 2e₂&minus;e₃, A to e₂&minus;e₃ and so on. In euclidean coordinates these vectors look as follows:
{|
|
:(1,&minus;1,0), (&minus;1,1,0)
:(1,0,&minus;1), (&minus;1,0,1)
:(0,1,&minus;1), (0,&minus;1,1)
|
:(2,&minus;1,&minus;1), (&minus;2,1,1)
:(1,&minus;2,1), (&minus;1,2,&minus;1)
:(1,1,&minus;2), (&minus;1,&minus;1,2)
|}

The corresponding set of '''simple roots''' is:
:e₁&minus;e₂ = (1,&minus;1,0), and &minus;e₁+2e₂&minus;e₃ = (&minus;1,2,&minus;1)
Note: α and A together form root system ''identical'' to [[Root_system#An|A₂]], while the system formed by β and B is ''isomorphic'' to [[Root_system#An|A₂]].

=== Weyl/Coxeter group ===
Its [[Weyl group|Weyl]]/[[Coxeter group|Coxeter]] group <math>G = W(G_2)</math> is the [[dihedral group]] <math>D_6</math> of [[Coxeter group#Properties|order]] 12. It has minimal faithful degree <math>\mu(G) = 5</math>.

=== Special holonomy ===
G<sub>2</sub> is one of the possible special groups that can appear as the [[holonomy]] group of a [[Riemannian metric]]. The [[manifold]]s of G<sub>2</sub> holonomy are also called [[G2 manifold|G<sub>2</sub>-manifolds]].

== Polynomial invariant==
G<sub>2</sub> is the automorphism group of the following two polynomials in 7 non-commutative variables.

:<math>C_1 = t^2+u^2+v^2+w^2+x^2+y^2+z^2</math>
:<math>C_2 = tuv + wtx  + ywu + zyt + vzw + xvy  + uxz  </math> (± permutations)

which comes from the octonion algebra. The variables must be non-commutative otherwise the second polynomial would be identically zero.

== Generators ==
Adding a representation of the 14 generators with coefficients ''A'',&nbsp;...,&nbsp;''N'' gives the matrix:
: <math>A\lambda_1+\cdots+N\lambda_{14}=
\begin{bmatrix}
 0 & C &-B & E &-D &-G &F-M \\
-C & 0 & A & F &-G+N&D-K&-E-L \\
 B &-A & 0 &-N & M & L & -K \\
-E &-F & N & 0 &-A+H&-B+I&C-J\\
 D &G-N &-M &A-H& 0 & J &I \\
 G &K-D& -L&B-I&-J & 0 & -H \\
-F+M&E+L& K &-C+J& -I & H & 0
\end{bmatrix}</math>

It is exactly the Lie algebra of the group
: <math>G_2=\{g\in \mathrm{SO}(7):g^*\varphi=\varphi, \varphi = \omega^{123} + \omega^{145} + \omega^{167} + \omega^{246} - \omega^{257} - \omega^{347} - \omega^{356}\}</math>

There are 480 different representations of <math>G_2</math> corresponding to the 480 representations of octonions. The calibrated form, <math>\varphi</math> has 30 different forms and each has 16 different signed variations. Each of the signed variations generate signed differences of <math>G_2</math> and each is an automorphism of all 16 corresponding octonions. Hence there are really only 30 different representations of <math>G_2</math>. These can all be constructed with Clifford algebra<ref>{{citation |url=https://github.com/GPWilmot/geoalg|title=Construction of G2 using Clifford Algebra|year=2023|last=Wilmot|first=G.P.}}</ref> using an invertible form <math>3e_{1234567}\pm\varphi</math> for octonions. For other signed variations of <math>\varphi</math>, this form has remainders that classify 6 other non-associative algebras that show partial <math>G_2</math> symmetry. An analogous calibration in <math>\mathrm{Spin}(15)</math> leads to sedenions and at least 11 other related algebras.

==Representations==
[[File:G2 Maximal Embeddings.svg|thumb|300px|Embeddings of the maximal subgroups of G<sub>2</sub> up to dimension 77 with associated projection matrix.]]

The characters of finite-dimensional representations of the real and complex Lie algebras and Lie groups are all given by the [[Weyl character formula]].  The dimensions of the smallest irreducible representations are {{OEIS|id=A104599}}:

:1, 7, 14, 27, 64, 77 (twice), 182, 189, 273, 286, 378, 448, 714, 729, 748, 896, 924, 1254, 1547, 1728, 1729, 2079 (twice), 2261, 2926, 3003, 3289, 3542, 4096, 4914, 4928 (twice), 5005, 5103, 6630, 7293, 7371, 7722, 8372, 9177, 9660, 10206, 10556, 11571, 11648, 12096, 13090....

The 14-dimensional representation is the [[Adjoint representation of a Lie algebra|adjoint representation]], and the 7-dimensional one is action of G<sub>2</sub> on the imaginary octonions.

There are two non-isomorphic irreducible representations of dimensions 77, 2079, 4928, 30107, etc. The [[fundamental representation]]s are those with dimensions 14 and 7 (corresponding to the two nodes in the [[#Dynkin diagram|Dynkin diagram]] in the order such that the triple arrow points from the first to the second).

{{harvtxt|Vogan|1994}} described the (infinite-dimensional) unitary irreducible representations of the split real form of G<sub>2</sub>.

The embeddings of the maximal subgroups of G<sub>2</sub> up to dimension 77 are shown to the right.

==Finite groups==
The group G<sub>2</sub>(''q'') is the points of the algebraic group G<sub>2</sub> over the [[finite field]] '''F'''<sub>''q''</sub>. These finite groups were first introduced by [[Leonard Eugene Dickson]] in {{harvtxt|Dickson|1901}} for odd ''q'' and {{harvtxt|Dickson|1905}} for even ''q''. The order of G<sub>2</sub>(''q'') is {{nowrap|''q''<sup>6</sup>(''q''<sup>6</sup> − 1)(''q''<sup>2</sup> − 1)}}. When {{nowrap|''q'' ≠ 2}}, the group is [[simple group|simple]], and when {{nowrap|1=''q'' = 2}}, it has a simple subgroup of [[Index of a subgroup|index]] 2 isomorphic to <sup>2</sup>''A''<sub>2</sub>(3<sup>2</sup>), and is the automorphism group of a maximal order of the octonions. The Janko group [[Janko group J1|J<sub>1</sub>]] was first constructed as a subgroup of G<sub>2</sub>(11). {{harvtxt|Ree|1960}} introduced twisted [[Ree group]]s <sup>2</sup>G<sub>2</sub>(''q'') of order {{nowrap|''q''<sup>3</sup>(''q''<sup>3</sup> + 1)(''q'' − 1)}} for {{nowrap|1=''q'' = 3<sup>2''n''+1</sup>}}, an odd power of 3.

==See also==
* [[Cartan matrix]]
* [[Dynkin diagram]]
* [[Exceptional Jordan algebra]]
* [[Fundamental representation]]
* [[G2-structure|G<sub>2</sub>-structure]]
* [[Lie group]]
* [[Seven-dimensional cross product]]
* [[Simple Lie group]]
* [[Star of David]]

==References==
{{Reflist}}
*{{Citation | last1=Adams | first1=J. Frank | title=Lectures on exceptional Lie groups | url=https://books.google.com/books?isbn=0226005275 | publisher=[[University of Chicago Press]] | series=Chicago Lectures in Mathematics | isbn=978-0-226-00526-3 | mr=1428422 | year=1996}}
* {{citation|first=John|last=Baez|author-link=John Baez|title=The Octonions| journal=Bull. Amer. Math. Soc.|volume=39|year=2002|pages=145–205|doi=10.1090/S0273-0979-01-00934-X|issue=2|arxiv=math/0105155|s2cid=586512 }}.
::See section 4.1: G<sub>2</sub>; an online HTML version of which is available at http://math.ucr.edu/home/baez/octonions/node14.html.
*{{Citation | last=Bryant|first=Robert|author-link=Robert Bryant (mathematician)|title=Metrics with Exceptional Holonomy|journal=Annals of Mathematics|year=1987|volume=126|series=2|issue=3|pages=525–576|doi=10.2307/1971360|jstor=1971360}} 
*{{Citation | last1=Dickson | first1=Leonard Eugene | author1-link=Leonard Eugene Dickson | title=Theory of Linear Groups in An Arbitrary Field | publisher=[[American Mathematical Society]] | location=Providence, R.I. | id=Reprinted in volume II of his collected papers | year=1901 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=2 | issue=4 | pages=363–394 | jstor=1986251 | doi=10.1090/S0002-9947-1901-1500573-3| doi-access=free }} Leonard E. Dickson reported groups of type G<sub>2</sub> in fields of odd characteristic.
*{{citation|author-link=L. E. Dickson|first=L. E.|last= Dickson|title=A new system of simple groups|journal=Math. Ann.|volume= 60 |year=1905|pages=137–150|doi=10.1007/BF01447497|s2cid=179178145 |url=https://zenodo.org/record/2475009}} Leonard E. Dickson reported groups of type G<sub>2</sub> in fields of even characteristic.
*{{Citation | last1=Ree | first1=Rimhak | title=A family of simple groups associated with the simple Lie algebra of type (G<sub>2</sub>) | doi=10.1090/S0002-9904-1960-10523-X  | mr=0125155 | year=1960 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=66 | pages=508–510 | issue=6| doi-access=free }}
*{{Citation | last1=Vogan | first1=David A. Jr. | title=The unitary dual of G<sub>2</sub> | doi=10.1007/BF01231578 | year=1994 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=116 | issue=1 | pages=677–791 | mr=1253210| bibcode=1994InMat.116..677V | s2cid=120845135 }}

{{Exceptional_Lie_groups}}
{{String theory topics |state=collapsed}}

[[Category:Algebraic groups]]
[[Category:Lie groups]]
[[Category:Octonions]]
[[Category:Exceptional Lie algebras]]