imported>Pratyush Sarkar: /* Algebra */ Corrected a mistake

2024-07-24T18:40:47Z

Algebra: Corrected a mistake

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{{Short description|Simple Lie group; the automorphism group of the octonions}}
{{DISPLAYTITLE:G2 (mathematics)}}
{{Group theory sidebar |Topological}}
{{Lie groups |Simple}}

In [[mathematics]], '''G2''' 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. G2 has rank 2 and dimension 14. It has two [[fundamental representation]]s, with dimension 7 and 14.

The compact form of G2 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''2
| 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 = G2 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 = G2 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 G2.<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 G2.<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, G2 is sometimes denoted by E2.

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

*The underlying real Lie algebra of the complex Lie algebra G2 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 G2.
*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''2 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 G2 ===
{| class=wikitable width=480
|- valign=top
|[[File:Root system G2.svg|160px]] The 12 vector [[root system]] of G2 in 2 dimensions.
|[[File:3-cube t1.svg|160px]] The A2 [[Coxeter plane]] projection of the 12 vertices of the [[cuboctahedron]] contain the same 2D vector arrangement.
|[[Image:G2Coxeter.svg|160px]] 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,−3) and (−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−e₂, β to −e₁ + 2e₂−e₃, A to e₂−e₃ and so on. In euclidean coordinates these vectors look as follows:
{|
|
:(1,−1,0), (−1,1,0)
:(1,0,−1), (−1,0,1)
:(0,1,−1), (0,−1,1)
|
:(2,−1,−1), (−2,1,1)
:(1,−2,1), (−1,2,−1)
:(1,1,−2), (−1,−1,2)
|}

The corresponding set of '''simple roots''' is:
:e₁−e₂ = (1,−1,0), and −e₁+2e₂−e₃ = (−1,2,−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 ===
G2 is one of the possible special groups that can appear as the [[holonomy]] group of a [[Riemannian metric]]. The [[manifold]]s of G2 holonomy are also called [[G2 manifold|G2-manifolds]].

== Polynomial invariant==
G2 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'', ..., ''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 G2 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 G2 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 G2.

The embeddings of the maximal subgroups of G2 up to dimension 77 are shown to the right.

==Finite groups==
The group G2(''q'') is the points of the algebraic group G2 over the [[finite field]] '''F'''''q''. 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 G2(''q'') is {{nowrap|''q''6(''q''6 − 1)(''q''2 − 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 2''A''2(32), and is the automorphism group of a maximal order of the octonions. The Janko group [[Janko group J1|J1]] was first constructed as a subgroup of G2(11). {{harvtxt|Ree|1960}} introduced twisted [[Ree group]]s 2G2(''q'') of order {{nowrap|''q''3(''q''3 + 1)(''q'' − 1)}} for {{nowrap|1=''q'' = 32''n''+1}}, an odd power of 3.

==See also==
* [[Cartan matrix]]
* [[Dynkin diagram]]
* [[Exceptional Jordan algebra]]
* [[Fundamental representation]]
* [[G2-structure|G2-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: G2; 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 G2 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 G2 in fields of even characteristic.
*{{Citation | last1=Ree | first1=Rimhak | title=A family of simple groups associated with the simple Lie algebra of type (G2) | 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 G2 | 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]]

G2 (mathematics) - Revision history

imported>Pratyush Sarkar: /* Algebra */ Corrected a mistake