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| {{DISPLAYTITLE:G<sub>2</sub> (mathematics)}}
| | Im Britt and was born on 21 November 1981. My hobbies are Videophilia (Home theater) and Herping.<br><br>Take a look at my web blog; [http://printerspace.com/michael.html michael kors outlet] |
| {{Group theory sidebar |Topological}}
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| {{Lie groups |Simple}}
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| In [[mathematics]], '''G<sub>2</sub>''' is the name of 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.
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| 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]]. [[Robert Bryant]] introduced the definition of G<sub>2</sub> as the subgroup of <math>GL(\mathbb{R}^7)</math> which preserves the non-degenerate 3-form
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| :<math>dx^{123}+dx^{145}+dx^{167}+dx^{246}-dx^{257}-dx^{347}-dx^{356},</math>
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| with <math>dx^{ijk}</math> denoting <math>dx^i\wedge dx^j\wedge dx^k.</math>
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| In older books and papers, G<sub>2</sub> is sometimes denoted by E<sub>2</sub>.
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| ==Real forms==
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| There are 3 simple real Lie algebras associated with this root system:
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| *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>.
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| *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.
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| *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 SU(2) × SU(2)/(−1,−1). It has a non-algebraic double cover that is simply connected.
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| == Algebra ==
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| ===Dynkin diagram and Cartan matrix ===
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| The [[Dynkin diagram]] for G<sub>2</sub> is given by [[Image:Dynkin diagram G2.png|Dynkin diagram of G_2]].
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| Its [[Cartan matrix]] is:
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| :<math>\left [\begin{smallmatrix}
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| \;\,\, 2&-3\\
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| -1&\;\,\, 2
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| \end{smallmatrix}\right ]</math>
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| === Roots of G<sub>2</sub> ===
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| {| class=wikitable width=480
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| |- valign=top
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| |[[File:Root system G2.svg|160px]]<BR>The 12 vector [[root system]] of G<sub>2</sub> in 2 dimensions.
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| |[[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.
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| |[[Image:G2Coxeter.svg|160px]]<BR>Graph of G2 as a subgroup of F4 and E8 projected into the Coxeter plane
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| |}
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| Although they [[Linear span|span]] a 2-dimensional space, as drawn, it's much more symmetric to consider them as [[Vector space|vectors]] in a 2-dimensional subspace of a three dimensional space.
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| {|
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| :(1,−1,0), (−1,1,0)
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| :(1,0,−1), (−1,0,1)
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| :(0,1,−1), (0,−1,1)
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| :(2,−1,−1), (−2,1,1)
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| :(1,−2,1), (−1,2,−1)
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| :(1,1,−2), (−1,−1,2)
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| |}
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| One set of '''simple roots''', for {{Dynkin2|node_n1|6a|node_n2}} is:
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| :(0,1,−1), (1,−2,1)
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| === Weyl/Coxeter group ===
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| Its [[Weyl group|Weyl]]/[[Coxeter group|Coxeter]] group is the [[dihedral group]], D<sub>6</sub> of [[Coxeter_group#Properties|order]] 12.
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| === Special holonomy ===
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| 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]].
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| == Polynomial Invariant==
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| G<sub>2</sub> is the automorphism group of the following two polynomials in 7 non-commutative variables.
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| :<math>C_1 = t^2+u^2+v^2+w^2+x^2+y^2+z^2</math>
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| :<math>C_2 = tuv + wtx + ywu + zyt + vzw + xvy + uxz </math> (± permutations)
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| which comes from the octionion algebra. The variables must be non-commutative otherwise the second polynomial would be identically zero.
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| ==Generators==
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| Adding a representation of the 14 generators with coefficients A..N gives the matrix:
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| :<math>A\lambda_1+...+N\lambda_{14}=
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| \begin{bmatrix}
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| 0 & C &-B & E &-D &-G &-F+M \\
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| -C & 0 & A & F &-G+N&D-K&E+L \\
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| B &-A & 0 &-N & M & L & K \\
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| -E &-F & N & 0 &-A+H&-B+I&-C+J\\
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| D &G-N &-M &A-H& 0 & J &-I \\
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| G &K-D& -L&B-I&-J & 0 & H \\
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| F-M&-E-L& -K &C-J& I & -H & 0 \\
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| \end{bmatrix}</math>
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| ==Representations==
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| 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}}:
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| :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….
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| 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.
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| There are two non-isomorphic irreducible representations of dimensions 77, 2079, 4928, 28652, 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).
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| {{harvtxt|Vogan|1994}} described the (infinite dimensional) unitary irreducible representations of the.split real form of G<sub>2</sub>.
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| ==Finite groups==
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| 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 ''q''≠2, the group is [[simple group|simple]], and when ''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>). The [[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 ''q''<sup>3</sup>(''q''<sup>3</sup>+1)(''q''−1) for ''q''=3<sup>2''n''+1</sup> an odd power of 3.
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| ==See also==
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| * [[Cartan matrix]]
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| * [[Dynkin diagram]]
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| * [[Exceptional Jordan algebra]]
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| * [[Fundamental representation]]
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| * [[Lie group]]
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| * [[Seven-dimensional cross product]]
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| * [[Simple Lie group]]
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| ==References==
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| *{{Citation | last1=Adams | first1=J. Frank | title=Lectures on exceptional Lie groups | url=http://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}}
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| * {{citation|first=Ilka|last=Agricola|url=http://www.ams.org/notices/200808/tx080800922p.pdf|title=Old and New on the Exceptional Group G<sub>2</sub>|year=2008|volume=55|issue=8}}
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| * {{citation|first=John|last=Baez|authorlink=John Baez|title=The Octonions|url=http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/home.html| journal=Bull. Amer. Math. Soc.|volume=39|year=2002|pages=145–205|doi=10.1090/S0273-0979-01-00934-X|issue=2}}.
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| ::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.
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| *{{Citation | last=Bryant|first=Robert|title=Metrics with Exceptional Holonomy|journal=Annals of Mathematics|year=1987|volume=126|series=2|issue=3|pages=525–576}}
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| *{{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}} Leonard E. Dickson reported groups of type G<sub>2</sub> in fields of odd characteristic.
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| *{{citation|authorlink=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}} Leonard E. Dickson reported groups of type G<sub>2</sub> in fields of even characteristic.
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| *{{Citation | last1=Ree | first1=Rimhak | title=A family of simple groups associated with the simple Lie algebra of type (G<sub>2</sub>) | url=http://www.ams.org/journals/bull/1960-66-06/S0002-9904-1960-10523-X/home.html | 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}}
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| *{{Citation | last1=Vogan | first1=David A. Jr. | title=The unitary dual of G<sub>2</sub> | url=http://dx.doi.org/10.1007/BF01231578 | doi=10.1007/BF01231578 | year=1994 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=116 | issue=1 | pages=677–791 | mr=1253210}}
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| {{Exceptional_Lie_groups}}
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| [[Category:Algebraic groups]]
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| [[Category:Lie groups]]
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| [[Category:Octonions]]
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