Editing G2 (mathematics) (section)

== Algebra ==

===Dynkin diagram and Cartan matrix ===
The [[Dynkin diagram]] for ''G''<sub>2</sub> is given by [[Image:Dynkin diagram G2.png|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
|[[File:Root system G2.svg|160px]]<BR>The 12 vector [[root system]] of G<sub>2</sub> in 2 dimensions.
|[[File:3-cube t1.svg|160px]]<BR>The A<sub>2</sub> [[Coxeter plane]] projection of the 12 vertices of the [[cuboctahedron]] contain the same 2D vector arrangement.
|[[Image:G2Coxeter.svg|160px]]<BR>Graph of G2 as a subgroup of F4 and E8 projected into the Coxeter plane
|}

A set of '''simple roots''' for {{Dynkin2|node_n1|6a|node_n2}} can be read directly from the Cartan matrix above. These are (2,&minus;3) and (&minus;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|(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 [[Linear span|span]] a 2-dimensional space, as drawn, it is much more symmetric to consider them as [[Vector space|vectors]] in a 2-dimensional subspace of a three-dimensional space. In this identification α corresponds to e₁&minus;e₂, β to &minus;e₁ + 2e₂&minus;e₃, A to e₂&minus;e₃ and so on. In euclidean coordinates these vectors look as follows:
{|
|
:(1,&minus;1,0), (&minus;1,1,0)
:(1,0,&minus;1), (&minus;1,0,1)
:(0,1,&minus;1), (0,&minus;1,1)
|
:(2,&minus;1,&minus;1), (&minus;2,1,1)
:(1,&minus;2,1), (&minus;1,2,&minus;1)
:(1,1,&minus;2), (&minus;1,&minus;1,2)
|}

The corresponding set of '''simple roots''' is:
:e₁&minus;e₂ = (1,&minus;1,0), and &minus;e₁+2e₂&minus;e₃ = (&minus;1,2,&minus;1)
Note: α and A together form root system ''identical'' to [[Root_system#An|A₂]], while the system formed by β and B is ''isomorphic'' to [[Root_system#An|A₂]].

=== Weyl/Coxeter group ===
Its [[Weyl group|Weyl]]/[[Coxeter group|Coxeter]] group <math>G = W(G_2)</math> is the [[dihedral group]] <math>D_6</math> of [[Coxeter group#Properties|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 [[manifold]]s of G<sub>2</sub> holonomy are also called [[G2 manifold|G<sub>2</sub>-manifolds]].