G2 (mathematics)

They are the smallest of the five exceptional simple Lie groups.

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).

, 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

[1] In 1893, Élie Cartan published a note describing an open set in

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

[2] In the same year, in the same journal, Engel noticed the same thing.

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.

[3][4] 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.

[6] In 1914 he stated that this is the compact real form of G2.

[7] In older books and papers, G2 is sometimes denoted by E2.

There are 3 simple real Lie algebras associated with this root system: The Dynkin diagram for G2 is given by .

These are (2,−3) and (−1, 2), however the integer lattice spanned by those is not the one pictured above (from obvious reason: the hexagonal lattice on the plane cannot be generated by integer vectors).

The diagram above is obtained from a different pair roots:

In euclidean coordinates these vectors look as follows: The corresponding set of simple roots is: Note: α and A together form root system identical to A₂, while the system formed by β and B is isomorphic to A₂.

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

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: It is exactly the Lie algebra of the group There are 480 different representations of

These can all be constructed with Clifford algebra[8] using an invertible form

, this form has remainders that classify 6 other non-associative algebras that show partial

leads to sedenions and at least 11 other related algebras.

The characters of finite-dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula.

There are two non-isomorphic irreducible representations of dimensions 77, 2079, 4928, 30107, 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.

The embeddings of the maximal subgroups of G2 up to dimension 77 are shown to the right.

When q ≠ 2, the group is simple, and when q = 2, it has a simple subgroup of index 2 isomorphic to 2A2(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).