G₂ (mathematics)

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. Its fundamental representation is 7-dimensional.

The compact form of G₂ can be described as the automorphism group of the octonion algebra or, equivalently, as the subgroup of that preserves any chosen particular vector in its 8-dimensional real spinor representation.

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 or

Its Cartan matrix is:

Roots of G₂

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.

Simple roots

(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

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

Leonard Eugene Dickson introduced the finite groups G₂(F_q) in Dickson (1901) for odd q and Dickson (1905) for even q. The group G₂(F_q) is the points of the algebraic group G₂ over the finite field F_q. The order of the group 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 Janko group J1 was first constructed as a subgroup of G₂(F₁₁). Ree (1960) introduced twisted Ree groups ²G₂(F_q) of order q³(q³+1)(q−1) for q=3²ⁿ⁺¹ an odd power of 3.

See also

• Seven-dimensional cross product

References

• Adams, J. Frank (1996), Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University of Chicago Press, ISBN 978-0-226-00526-3, MR1428422, http://books.google.com/books?isbn=0226005275 
• Agricola, Ilka (2008), Old and New on the Exceptional Group G₂, 55, http://www.ams.org/notices/200808/tx080800922p.pdf 
• Baez, John (2002), "The Octonions", Bull. Amer. Math. Soc. 39 (2): 145–205, doi:10.1090/S0273-0979-01-00934-X, http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/home.html .

See section 4.1: G₂; an online HTML version of which is available at http://math.ucr.edu/home/baez/octonions/node14.html.

• Dickson, Leonard Eugene (1901), "Theory of Linear Groups in An Arbitrary Field" (in English), 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, MR0125155, http://www.ams.org/journals/bull/1960-66-06/S0002-9904-1960-10523-X/home.html 
• Vogan, David A. Jr. (1994), "The unitary dual of G₂", Inventiones Mathematicae 116 (1): 677–791, doi:10.1007/BF01231578, ISSN 0020-9910, MR1253210, http://dx.doi.org/10.1007/BF01231578