G2 (mathematics)

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In mathematics, G₂ is the name of some Lie groups and also their Lie algebras [\mathfrak_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.

Contents

1 Algebra

1.1
1.2 Roots of G₂
1.3 Weyl/Coxeter group
1.4
1.5 Special holonomy

2 References

Algebra

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.

[\begin2&-3\\-1&2\end]

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.

Exceptional Lie groups
E₆ | E₇ | E_7½ | E₈ | F₄ | G₂

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