G_{2} (mathematics)
Algebraic structure → Group theory Group theory 

Group theory → Lie groups Lie groups 

In mathematics, G_{2} is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras [math]\displaystyle{ \mathfrak{g}_2, }[/math] as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups. G_{2} has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14.
The compact form of G_{2} can be described as the automorphism group of the octonion algebra or, equivalently, as the subgroup of SO(7) that preserves any chosen particular vector in its 8dimensional real spinor representation (a spin representation).
History
The Lie algebra [math]\displaystyle{ \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 saying that he had found a 14dimensional simple Lie algebra, which we now call [math]\displaystyle{ \mathfrak{g}_2 }[/math].^{[1]}
In 1893, Élie Cartan published a note describing an open set in [math]\displaystyle{ \mathbb{C}^5 }[/math] equipped with a 2dimensional distribution—that is, a smoothly varying field of 2dimensional subspaces of the tangent space—for which the Lie algebra [math]\displaystyle{ \mathfrak{g}_2 }[/math] appears as the infinitesimal symmetries.^{[2]} In the same year, in the same journal, Engel noticed the same thing. Later it was discovered that the 2dimensional distribution is closely related to a ball rolling on another ball. The space of configurations of the rolling ball is 5dimensional, with a 2dimensional distribution that describes motions of the ball where it rolls without slipping or twisting.^{[3]}^{[4]}
In 1900, Engel discovered that a generic antisymmetric trilinear form (or 3form) on a 7dimensional complex vector space is preserved by a group isomorphic to the complex form of G_{2}.^{[5]}
In 1908 Cartan mentioned that the automorphism group of the octonions is a 14dimensional simple Lie group.^{[6]} In 1914 he stated that this is the compact real form of G_{2}.^{[7]}
In older books and papers, G_{2} is sometimes denoted by E_{2}.
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_{2} 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_{2}.
 The Lie algebra of the compact form is 14dimensional. The associated Lie group has no outer automorphisms, no center, and is simply connected and compact.
 The Lie algebra of the noncompact (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 nonalgebraic double cover that is simply connected.
Algebra
Dynkin diagram and Cartan matrix
The Dynkin diagram for G_{2} is given by .
Its Cartan matrix is:
 [math]\displaystyle{ \left [\begin{array}{rr} 2 & 3 \\ 1 & 2 \end{array}\right] }[/math]
Roots of G_{2}
The 12 vector root system of G_{2} in 2 dimensions. 
The A_{2} Coxeter plane projection of the 12 vertices of the cuboctahedron contain the same 2D vector arrangement. 
Graph of G2 as a subgroup of F4 and E8 projected into the Coxeter plane 
A set of simple roots for 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]\displaystyle{ \alpha = \left( \sqrt{2}, 0 \right) }[/math] and [math]\displaystyle{ \beta = \left(\sqrt{2}\cos{\frac{5\pi}{6}},\sin{\frac{5\pi}{6}}\right) = \frac{1}{2}\left(\sqrt{6},1 \right) }[/math]. The remaining (positive) roots are A = α + β, B = 3α + β, α + A = 2α + β, and A + B = 3α + 2β. Although they do span a 2dimensional space, as drawn, it is much more symmetric to consider them as vectors in a 2dimensional subspace of a threedimensional 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:


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 A₂, while the system formed by β and B is isomorphic to A₂.
Weyl/Coxeter group
Its Weyl/Coxeter group [math]\displaystyle{ G = W(G_2) }[/math] is the dihedral group [math]\displaystyle{ D_6 }[/math] of order 12. It has minimal faithful degree [math]\displaystyle{ \mu(G) = 5 }[/math].
Special holonomy
G_{2} is one of the possible special groups that can appear as the holonomy group of a Riemannian metric. The manifolds of G_{2} holonomy are also called G_{2}manifolds.
Polynomial invariant
G_{2} is the automorphism group of the following two polynomials in 7 noncommutative variables.
 [math]\displaystyle{ C_1 = t^2+u^2+v^2+w^2+x^2+y^2+z^2 }[/math]
 [math]\displaystyle{ C_2 = tuv + wtx + ywu + zyt + vzw + xvy + uxz }[/math] (± permutations)
which comes from the octonion algebra. The variables must be noncommutative otherwise the second polynomial would be identically zero.
Generators
Adding a representation of the 14 generators with coefficients A, ..., N gives the matrix:
 [math]\displaystyle{ A\lambda_1+\cdots+N\lambda_{14}= \begin{bmatrix} 0 & C &B & E &D &G &FM \\ C & 0 & A & F &G+N&DK&EL \\ B &A & 0 &N & M & L & K \\ E &F & N & 0 &A+H&B+I&CJ\\ D &GN &M &AH& 0 & J &I \\ G &KD& L&BI&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]\displaystyle{ G_2=\{g\in SO(7):g^*\varphi=\varphi, \varphi = \omega^{123} + \omega^{145} + \omega^{167} + \omega^{246}  \omega^{257}  \omega^{347}  \omega^{356}\} }[/math]
Representations
The characters of finitedimensional 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 (sequence A104599 in the OEIS):
 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 14dimensional representation is the adjoint representation, and the 7dimensional one is action of G_{2} on the imaginary octonions.
There are two nonisomorphic irreducible representations of dimensions 77, 2079, 4928, 30107, etc. The fundamental representations are those with dimensions 14 and 7 (corresponding to the two nodes in the Dynkin diagram in the order such that the triple arrow points from the first to the second).
(Vogan 1994) described the (infinitedimensional) unitary irreducible representations of the split real form of G_{2}.
Finite groups
The group G_{2}(q) is the points of the algebraic group G_{2} over the finite field F_{q}. These finite groups were first introduced by Leonard Eugene Dickson in (Dickson 1901) for odd q and (Dickson 1905) for even q. The order of G_{2}(q) is q^{6}(q^{6} − 1)(q^{2} − 1). When q ≠ 2, the group is simple, and when q = 2, it has a simple subgroup of index 2 isomorphic to ^{2}A_{2}(3^{2}), and is the automorphism group of a maximal order of the octonions. The Janko group J_{1} was first constructed as a subgroup of G_{2}(11). (Ree 1960) introduced twisted Ree groups ^{2}G_{2}(q) of order q^{3}(q^{3} + 1)(q − 1) for q = 3^{2n+1}, an odd power of 3.
See also
 Cartan matrix
 Dynkin diagram
 Exceptional Jordan algebra
 Fundamental representation
 G_{2}structure
 Lie group
 Sevendimensional cross product
 Simple Lie group
References
 ↑ "Old and new on the exceptional group G_{2}". Notices of the American Mathematical Society 55 (8): 922–929. 2008. https://www.ams.org/notices/200808/tx080800922p.pdf.
 ↑ Élie Cartan (1893). "Sur la structure des groupes simples finis et continus". C. R. Acad. Sci. 116: 784–786.
 ↑ Gil Bor and Richard Montgomery (2009). "G_{2} and the "rolling distribution"". L'Enseignement Mathématique 55: 157–196. doi:10.4171/lem/5518.
 ↑ John Baez and John Huerta (2014). "G_{2} and the rolling ball". Trans. Amer. Math. Soc. 366 (10): 5257–5293. doi:10.1090/s000299472014059771.
 ↑ Friedrich Engel (1900). "Ein neues, dem linearen Komplexe analoges Gebilde". Leipz. Ber. 52: 63–76,220–239.
 ↑ Élie Cartan (1908). "Nombres complexes". Encyclopedie des Sciences Mathematiques. Paris: GauthierVillars. pp. 329–468.
 ↑ Élie Cartan (1914), "Les groupes reels simples finis et continus", Ann. Sci. École Norm. Sup. 31: 255–262
 Adams, J. Frank (1996), Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University of Chicago Press, ISBN 9780226005263, https://books.google.com/books?isbn=0226005275
 Baez, John (2002), "The Octonions", Bull. Amer. Math. Soc. 39 (2): 145–205, doi:10.1090/S027309790100934X.
 See section 4.1: G_{2}; an online HTML version of which is available at http://math.ucr.edu/home/baez/octonions/node14.html.
 Bryant, Robert (1987), "Metrics with Exceptional Holonomy", Annals of Mathematics, 2 126 (3): 525–576, doi:10.2307/1971360
 Dickson, Leonard Eugene (1901), "Theory of Linear Groups in An Arbitrary Field", Transactions of the American Mathematical Society (Providence, R.I.: American Mathematical Society) 2 (4): 363–394, doi:10.1090/S00029947190115005733, Reprinted in volume II of his collected papers, ISSN 00029947 Leonard E. Dickson reported groups of type G_{2} in fields of odd characteristic.
 Dickson, L. E. (1905), "A new system of simple groups", Math. Ann. 60: 137–150, doi:10.1007/BF01447497, https://zenodo.org/record/2475009 Leonard E. Dickson reported groups of type G_{2} in fields of even characteristic.
 Ree, Rimhak (1960), "A family of simple groups associated with the simple Lie algebra of type (G_{2})", Bulletin of the American Mathematical Society 66 (6): 508–510, doi:10.1090/S00029904196010523X, ISSN 00029904
 Vogan, David A. Jr. (1994), "The unitary dual of G_{2}", Inventiones Mathematicae 116 (1): 677–791, doi:10.1007/BF01231578, ISSN 00209910, Bibcode: 1994InMat.116..677V
Original source: https://en.wikipedia.org/wiki/G2 (mathematics).
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