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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 \mathfrak{g}_2, 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 8-dimensional real spinor representation. Robert Bryant introduced the definition of G_{2} as the subgroup of \operatorname{GL}(\mathbb{R}^7) that preserves the non-degenerate 3-form
(invariant under the cyclic permutation (0123456)) with dx^{ijk} denoting dx^i\wedge dx^j\wedge dx^k.
In older books and papers, G_{2} is sometimes denoted by E_{2}.
There are 3 simple real Lie algebras associated with this root system:
The Dynkin diagram for G_{2} is given by .
Its Cartan matrix is:
One set of simple roots, for is:
Its Weyl/Coxeter group is the dihedral group, D_{6} of order 12.
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.
G_{2} is the automorphism group of the following two polynomials in 7 non-commutative variables.
which comes from the octonion algebra. The variables must be non-commutative otherwise the second polynomial would be identically zero.
Adding a representation of the 14 generators with coefficients A..N gives the 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 (sequence A104599 in OEIS):
The 14-dimensional representation is the adjoint representation, and the 7-dimensional one is action of G_{2} on the imaginary octonions.
There are two non-isomorphic 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).
Vogan (1994) described the (infinite-dimensional) unitary irreducible representations of the split real form of G_{2}.
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.
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