G2 (mathematics)

In mathematics, G2 is the name of a Lie group and also its Lie algebra <math>\mathfrak{g}_2<math>. 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 <math>SO(7)<math> that preserves any chosen particular vector in its 8-dimensional real spinor representation.

Contents

Algebra

Dynkin diagram

Dynkin diagram of G_2

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, D6.

Cartan matrix

<math>

\begin{pmatrix} 2&-3\\ -1&2 \end{pmatrix} <math>

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.

Exceptional Lie groups

E6 | E7 | E8 | F4 | G2
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