G2 (mathematics)

In mathematics, G₂ is three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras $\mathfrak_2,$ 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 $\mathfrak_2$, 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 $\mathfrak_2$.

In 1893, Élie Cartan published a note describing an open set in $\mathbb^5$ 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 $\mathfrak_2$ 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:

*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 . It has a non-algebraic double cover that is simply connected.

Algebra

Dynkin diagram and Cartan matrix

The Dynkin diagram for ''G''₂ is given by .

Its Cartan matrix is:

:
\left begin
2 & -3 \\
-1 & 2
\end\right

Roots of G₂

A set of simple roots for can be read directly from the Cartan matrix above. 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: $\alpha = \left( 1, 0 \right)$ and $\beta = \sqrt\left(\cos,\sin\right) = \frac\left(-3,\sqrt \right)$.

The remaining (positive) roots are $A = \alpha + \beta,\, B = 3\alpha + \beta,\, \alpha + A = 2\alpha + \beta \,\,\,\, \beta + B = 3\alpha + 2\beta$.

Although they do 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. In this identification α corresponds to e₁−e₂, β to −e₁ + 2e₂−e₃, A to e₂−e₃ and so on. In euclidean coordinates these vectors look as follows:

The corresponding set of simple roots is:
:e₁−e₂ = (1,−1,0), and −e₁+2e₂−e₃ = (−1,2,−1)
Note: α and A together form root system ''identical'' to A₂, while the system formed by β and B is ''isomorphic'' to A₂.

Weyl/Coxeter group

Its Weyl/ Coxeter group $G = W(G_2)$ is the dihedral group $D_6$ of order 12. It has minimal faithful degree $\mu(G) = 5$.

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.

:$C_1 = t^2+u^2+v^2+w^2+x^2+y^2+z^2$
:$C_2 = tuv + wtx + ywu + zyt + vzw + xvy + uxz$ (± permutations)

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:
: $A\lambda_1+\cdots+N\lambda_= \begin 0 & C &-B & E &-D &-G &F-M \\ -C & 0 & A & F &-G+N&D-K&-E-L \\ B &-A & 0 &-N & M & L & -K \\ -E &-F & N & 0 &-A+H&-B+I&C-J\\ D &G-N &-M &A-H& 0 & J &I \\ G &K-D& -L&B-I&-J & 0 & -H \\ -F+M&E+L& K &-C+J& -I & H & 0 \end$

It is exactly the Lie algebra of the group
: $G_2=\$

There are 480 different representations of $G_2$ corresponding to the 480 representations of octonions. The calibrated form, $\varphi$ has 30 different forms and each has 16 different signed variations. Each of the signed variations generate signed differences of $G_2$ and each is an automorphism of all 16 corresponding octonions. Hence there are really only 30 different representations of $G_2$. These can all be constructed with Clifford algebra using an invertible form $3e_\pm\varphi$ for octonions. For other signed variations of $\varphi$, this form has remainders that classify 6 other non-associative algebras that show partial $G_2$ symmetry. An analogous calibration in $\mathrm(15)$ leads to sedenions and at least 11 other related algebras.

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 :

: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, 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).

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.

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.

See also

* Cartan matrix
* Dynkin diagram
* Exceptional Jordan algebra
* Fundamental representation
* G₂-structure
* Lie group
* Seven-dimensional cross product
* Simple Lie group
* Star of David

References

*
* .
::See section 4.1: G₂; an online HTML version of which is available at http://math.ucr.edu/home/baez/octonions/node14.html.
*
* Leonard E. Dickson reported groups of type G₂ in fields of odd characteristic.
* Leonard E. Dickson reported groups of type G₂ in fields of even characteristic.
*
*

{{String theory topics , state=collapsed

Algebraic groups
Lie groups
Octonions
Exceptional Lie algebras