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 formula_1 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 (a spin representation).
History.
The Lie algebra formula_2, 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 14-dimensional simple Lie algebra, which we now call formula_2.
In 1893, Élie Cartan published a note describing an open set in formula_4 equipped with a 2-dimensional distribution—that is, a smoothly varying field of 2-dimensional subspaces of the tangent space—for which the Lie algebra formula_2 appears as the infinitesimal symmetries. In the same year, in the same journal, Engel noticed the same thing. Later it was discovered that the 2-dimensional distribution is closely related to a ball rolling on another ball. The space of configurations of the rolling ball is 5-dimensional, with a 2-dimensional distribution that describes motions of the ball where it rolls without slipping or twisting.
In 1900, Engel discovered that a generic antisymmetric trilinear form (or 3-form) on a 7-dimensional complex vector space is preserved by a group isomorphic to the complex form of G2.
In 1908 Cartan mentioned that the automorphism group of the octonions is a 14-dimensional simple Lie group. In 1914 he stated that this is the compact real form of G2.
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:
Algebra.
Dynkin diagram and Cartan matrix.
The Dynkin diagram for "G"2 is given by .
Its Cartan matrix is:
Roots of G2.
Although they span a 2-dimensional space, as drawn, it is 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:
Weyl/Coxeter group.
Its Weyl/Coxeter group formula_7 is the dihedral group, formula_8 of order 12. It has minimal faithful degree formula_9.
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.
which comes from the octonion 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:
It is exactly the Lie algebra of the group
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 :
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).
 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 for odd "q" and for even "q". The order of G2("q") is . When , the group is simple, and when , it has a simple subgroup of index 2 isomorphic to 2"A"2(32), and is the automorphism group of a maximal order of the octonions. The Janko group J1 was first constructed as a subgroup of G2(11). introduced twisted Ree groups 2G2("q") of order for , an odd power of 3.