G2 (mathematics)

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:

${\displaystyle \left[\begin{array}{rr}2& -3\\ -1& 2\end{array}\right]}$

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: ${\displaystyle \alpha =\left(1,0\right)}$ and $\beta =\sqrt{3}\left(\cos \frac{5\pi }{6},\sin \frac{5\pi }{6}\right)=\frac{1}{2}\left(-3,\sqrt{3}\right)$.

The remaining (positive) roots are $A=\alpha +\beta ,B=3\alpha +\beta ,\alpha +A=2\alpha +\beta \mathrm{a}\mathrm{n}\mathrm{d}\beta +B=3\alpha +2\beta$.

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 ${\displaystyle G=W(G_2)}$ is the dihedral group ${\displaystyle D_6}$ of order 12. It has minimal faithful degree ${\displaystyle \mu (G)=5}$.

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.

${\displaystyle C_1=t^2+u^2+v^2+w^2+x^2+y^2+z^2}$

${\displaystyle C_2=tuv+wtx+ywu+zyt+vzw+xvy+uxz}$ (± 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:

${\displaystyle A{\lambda }_1+\cdots +N{\lambda }_{14}=\left[\begin{array}{ccccccc}0& C& -B& E& -D& -G& F-M\\ -C& 0& A& F& -G+N& D-K& -E-L\\ B& -A& 0& -N& M& L& -K\\ -E& -F& N& 0& -A+H& -B+I& C-J\\ D& G-N& -M& A-H& 0& J& I\\ G& K-D& -L& B-I& -J& 0& -H\\ -F+M& E+L& K& -C+J& -I& H& 0\end{array}\right]}$

It is exactly the Lie algebra of the group

${\displaystyle G_2=\{g\in \mathrm{S}\mathrm{O}(7):g^{\ast }\phi =\phi ,\phi ={\omega }^{123}+{\omega }^{145}+{\omega }^{167}+{\omega }^{246}-{\omega }^{257}-{\omega }^{347}-{\omega }^{356}\}}$

There are 480 different representations of ${\displaystyle G_2}$ corresponding to the 480 representations of octonions. The calibrated form, ${\displaystyle \phi }$ has 30 different forms and each has 16 different signed variations. Each of the signed variations generate signed differences of ${\displaystyle G_2}$ and each is an automorphism of all 16 corresponding octonions. Hence there are really only 30 different representations of ${\displaystyle G_2}$. These can all be constructed with Clifford algebra using an invertible form ${\displaystyle 3e_{1234567}\pm \phi }$ for octonions. For other signed variations of ${\displaystyle \phi }$, this form has remainders that classify 6 other non-associative algebras that show partial ${\displaystyle G_2}$ symmetry. An analogous calibration in ${\displaystyle \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(15)}$ 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 (sequence A104599 in the 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, 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.

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), 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). Ree (1960) introduced twisted Ree groups ²G₂(q) of order q³(q³ + 1)(q − 1) for q = 3²ⁿ⁺¹, an odd power of 3.