G2 (mathematics)

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In mathematics, G₂ 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. G₂ has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14.

The compact form of G₂ 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 G₂ as the subgroup of which preserves the non-degenerate 3-form

with denoting

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 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 G₂ is given by .

Its Cartan matrix is:

Roots of G₂

The 12 vector root system of G2 in 2 dimensions.

The A2 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 2-dimensional space, as drawn, it's much more symmetric to consider them as vectors in a 2-dimensional subspace of a three dimensional space.

One set of simple roots, for is:

(0,1,−1), (1,−2,1)

Weyl/Coxeter group

Its Weyl/Coxeter group is the dihedral group, D₆ of order 12.

Special holonomy

G₂ is one of the possible special groups that can appear as the holonomy group of a Riemannian metric. The manifolds of G₂ holonomy are also called G₂-manifolds.

Polynomial Invariant

G₂ is the automorphism group of the following two polynomials in 7 non-commutative variables.

(± permutations)

which comes from the octionion 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:

Representations

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):

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, 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, 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 G₂.

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 Dickson (1901) for odd q and Dickson (1905) for even q. The order of G₂(q) is q⁶(q⁶−1)(q²−1). When q≠2, the group is simple, and when q = 2, it has a simple subgroup of index 2 isomorphic to ²A₂(3²). The J₁ was first constructed as a subgroup of G₂(11). Ree (1960) introduced twisted Ree groups ²G₂(q) of order q³(q³+1)(q−1) for q=3²ⁿ⁺¹ an odd power of 3.

See also

• Cartan matrix
• Dynkin diagram
• Exceptional Jordan algebra
• Fundamental representation
• Lie group
• Seven-dimensional cross product
• Simple Lie group

References

• Adams, J. Frank (1996), Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University of Chicago Press, ISBN 978-0-226-00526-3, MR 1428422 
• Agricola, Ilka (2008), Old and New on the Exceptional Group G₂ 55 (8) 
• Baez, John (2002), "The Octonions", Bull. Amer. Math. Soc. 39 (2): 145–205, doi:10.1090/S0273-0979-01-00934-X .

See section 4.1: G₂; 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 
• 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/S0002-9947-1901-1500573-3, ISSN 0002-9947, JSTOR 1986251, Reprinted in volume II of his collected papers  Leonard E. Dickson reported groups of type G₂ in fields of odd characteristic.
• Dickson, L. E. (1905), "A new system of simple groups", Math. Ann. 60: 137–150, doi:10.1007/BF01447497  Leonard E. Dickson reported groups of type G₂ in fields of even characteristic.
• Ree, Rimhak (1960), "A family of simple groups associated with the simple Lie algebra of type (G₂)", Bulletin of the American Mathematical Society 66 (6): 508–510, doi:10.1090/S0002-9904-1960-10523-X, ISSN 0002-9904, MR 0125155 
• Vogan, David A. Jr. (1994), "The unitary dual of G₂", Inventiones Mathematicae 116 (1): 677–791, doi:10.1007/BF01231578, ISSN 0020-9910, MR 1253210