In mathematics, G<sub>2</sub> is three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras <math>\mathfrak{g}_2,</math> as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups. G<sub>2</sub> has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14.
The compact form of G<sub>2</sub> 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 <math>\mathfrak{g}_2</math>, 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 <math>\mathfrak{g}_2</math>.
In 1893, Élie Cartan published a note describing an open set in <math>\mathbb{C}^5</math> 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 <math>\mathfrak{g}_2</math> 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 G<sub>2</sub>.
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 G<sub>2</sub>.
In older books and papers, G<sub>2</sub> is sometimes denoted by E<sub>2</sub>.
Real forms
There are 3 simple real Lie algebras associated with this root system:
- The underlying real Lie algebra of the complex Lie algebra G<sub>2</sub> 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 G<sub>2</sub>.
- 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<sub>2</sub> is given by Dynkin diagram of G 2.
Its Cartan matrix is:
: <math>
\left [\begin{array}{rr}
2 & -3 \\
-1 & 2
\end{array}\right]
</math>
Roots of G<sub>2</sub>
{| class=wikitable width=480
|- valign=top
|160px<BR>The 12 vector root system of G<sub>2</sub> in 2 dimensions.
|160px<BR>The A<sub>2</sub> Coxeter plane projection of the 12 vertices of the cuboctahedron contain the same 2D vector arrangement.
|160px<BR>Graph of G2 as a subgroup of F4 and E8 projected into the Coxeter plane
|}
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: <math>\alpha = \left( 1, 0 \right)</math> and <math display="inline">\beta = \sqrt{3}\left(\cos{\frac{5\pi}{6,\sin{\frac{5\pi}{6\right) = \frac{1}{2}\left(-3,\sqrt{3} \right)</math>.
The remaining (positive) roots are <math display="inline">A = \alpha + \beta,\, B = 3\alpha + \beta,\, \alpha + A = 2\alpha + \beta \,\,{\rm and }\,\, \beta + B = 3\alpha + 2\beta</math>.
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:
{|
|
:(1,−1,0), (−1,1,0)
:(1,0,−1), (−1,0,1)
:(0,1,−1), (0,−1,1)
|
:(2,−1,−1), (−2,1,1)
:(1,−2,1), (−1,2,−1)
:(1,1,−2), (−1,−1,2)
|}
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 <math>G = W(G_2)</math> is the dihedral group <math>D_6</math> of order 12. It has minimal faithful degree <math>\mu(G) = 5</math>.
Special holonomy
G<sub>2</sub> is one of the possible special groups that can appear as the holonomy group of a Riemannian metric. The manifolds of G<sub>2</sub> holonomy are also called G<sub>2</sub>-manifolds.
Polynomial invariant
G<sub>2</sub> is the automorphism group of the following two polynomials in 7 non-commutative variables.
:<math>C_1 = t^2+u^2+v^2+w^2+x^2+y^2+z^2</math>
:<math>C_2 = tuv + wtx + ywu + zyt + vzw + xvy + uxz </math> (± 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:
: <math>A\lambda_1+\cdots+N\lambda_{14}=
\begin{bmatrix}
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{bmatrix}</math>
It is exactly the Lie algebra of the group
: <math>G_2=\{g\in \mathrm{SO}(7):g^*\varphi=\varphi, \varphi = \omega^{123} + \omega^{145} + \omega^{167} + \omega^{246} - \omega^{257} - \omega^{347} - \omega^{356}\}</math>
There are 480 different representations of <math>G_2</math> corresponding to the 480 representations of octonions. The calibrated form, <math>\varphi</math> has 30 different forms and each has 16 different signed variations. Each of the signed variations generate signed differences of <math>G_2</math> and each is an automorphism of all 16 corresponding octonions. Hence there are really only 30 different representations of <math>G_2</math>. These can all be constructed with Clifford algebra using an invertible form <math>3e_{1234567}\pm\varphi</math> for octonions. For other signed variations of <math>\varphi</math>, this form has remainders that classify 6 other non-associative algebras that show partial <math>G_2</math> symmetry. An analogous calibration in <math>\mathrm{Spin}(15)</math> leads to sedenions and at least 11 other related algebras.
Representations
thumb|300px|Embeddings of the maximal subgroups of G<sub>2</sub> up to dimension 77 with associated projection matrix.
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 G<sub>2</sub> 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 G<sub>2</sub>.
The embeddings of the maximal subgroups of G<sub>2</sub> up to dimension 77 are shown to the right.
Finite groups
The group G<sub>2</sub>(q) is the points of the algebraic group G<sub>2</sub> over the finite field F<sub>q</sub>. These finite groups were first introduced by Leonard Eugene Dickson in for odd q and for even q. The order of G<sub>2</sub>(q) is . When , the group is simple, and when , it has a simple subgroup of index 2 isomorphic to <sup>2</sup>A<sub>2</sub>(3<sup>2</sup>), and is the automorphism group of a maximal order of the octonions. The Janko group J<sub>1</sub> was first constructed as a subgroup of G<sub>2</sub>(11). introduced twisted Ree groups <sup>2</sup>G<sub>2</sub>(q) of order for , an odd power of 3.
See also
- Cartan matrix
- Dynkin diagram
- Exceptional Jordan algebra
- Fundamental representation
- G<sub>2</sub>-structure
- Lie group
- Seven-dimensional cross product
- Simple Lie group
- Star of David
References
- .
::See section 4.1: G<sub>2</sub>; 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 G<sub>2</sub> in fields of odd characteristic.
- Leonard E. Dickson reported groups of type G<sub>2</sub> in fields of even characteristic.