In mathematics, G₂ is the name of a Lie group and also its Lie algebra . It is the smallest of the five exceptional simple Lie groups. G₂ has rank 2 and dimension 14. Its center is the trivial subgroup. Its outer automorphism group is the trivial group. Its fundamental representation is 7-dimensional.

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

It's Weyl/Coxeter group is the dihedral group, D₆.

Cartan matrix

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.