G2

Group theory

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In mathematics, G₂ is the name of three simple Lie groups (a complex form, a compact real form and a split real form) and of their Lie algebras . They are the smallest of the five exceptional simple Lie groups. G₂ has rank 2 and dimension 14. 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.

Real forms

There are 3 simple real Lie algebras associated with this root system:

• The underlying real Lie algebra of the complex Lie algebra G₂ has dimension 28. It has complex conjugation as an outer automorphism and is simply connected. The maximal compact subgroup of its associated group is the compact form of G₂.
• The Lie algebra of the compact form is 14 dimensional. The associated Lie group has no outer automorphisms, no center, and is simply connected and compact.
• The Lie algebra of the non-compact (split) form has dimension 14. The associated simple Lie group has fundamental group of order 2 and its outer automorphism group is the trivial group. Its maximal compact subgroup is SU(2)×SU(2)/(−1×−1). It has a non-algebraic double cover that is simply connected.

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, D₆ of order 12.

Cartan matrix

Special holonomy

G₂ is one of the possible special groups that can appear as the holonomy group of a Riemannian metric. The manifolds of G₂ holonomy are also called G₂-manifolds.

References

• Agricola, Ilka (2008), Old and New on the Exceptional Group G₂, 55, http://www.ams.org/notices/200808/tx080800922p.pdf 
• Baez, John (2002), "The Octonions", Bull. Amer. Math. Soc. 39: 145–205, doi:10.1090/S0273-0979-01-00934-X, http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/home.html .

See section 4.1: G₂; an online HTML version of which is available at http://math.ucr.edu/home/baez/octonions/node14.html.