G_{2} (mathematics)
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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 , 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. Robert Bryant introduced the definition of G_{2} as the subgroup of that preserves the nondegenerate 3form
(invariant under the cyclic permutation (0123456)) with denoting
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:
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 
Although they span a 2dimensional space, as drawn, it is much more symmetric to consider them as vectors in a 2dimensional subspace of a threedimensional space.


One set of simple roots, for Template:Dynkin2 is:
 (0,1,−1), (1,−2,1)
Weyl/Coxeter group
Its Weyl/Coxeter group is the dihedral group, D_{6} of order 12.
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.
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:
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 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, 28652, 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).
Template:Harvtxt 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 Template:Harvtxt for odd q and Template:Harvtxt 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). Template:Harvtxt 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
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 See section 4.1: G_{2}; an online HTML version of which is available at http://math.ucr.edu/home/baez/octonions/node14.html.
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CitationClass=citation }} Leonard E. Dickson reported groups of type G_{2} in fields of odd characteristic.
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CitationClass=citation }} Leonard E. Dickson reported groups of type G_{2} in fields of even characteristic.
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