G2 (mathematics)

In mathematics, G₂ is the name of some Lie groups and also their Lie algebras ${\mathfrak {g}}_{2}$. They are the smallest of the five exceptional simple Lie groups. G₂ has rank 2 and dimension 14. The compact form is simply connected, and the non-compact (split) form has fundamental group of order 2. Its outer automorphism group is the trivial group. Its fundamental representation is 7-dimensional.

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.

Algebra

Dynkin diagram

Roots of G₂

Although they span a 2-dimensional space, it's much more symmetric to consider them as vectors in a 2-dimensional subspace of a three dimensional space.

(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)

Simple roots

(0,1,−1), (1,−2,1)

Weyl/Coxeter group

Its Weyl/Coxeter group is the dihedral group, D₁₂ of order 12.

Cartan matrix

${\begin{pmatrix}2&-3\\-1&2\end{pmatrix}}$

Special holonomy

G₂ is one of the possible special groups that can appear as holonomy. The manifolds of G₂ holonomy are also called Joyce manifolds.

References

• John Baez, The Octonions, Section 4.1: G₂, Bull. Amer. Math. Soc. 39 (2002), 145-205. Online HTML version at

http://math.ucr.edu/home/baez/octonions/node14.html.