G₂

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. "G"₂ has rank 2 and dimension 14. Its fundamental representation is 7-dimensional.

The compact form of "G"₂ 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 "G"₂ 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 "G"₂.
*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, "D"₆ 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

* John Baez, "The Octonions", Section 4.1: G₂, [http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/home.html Bull. Amer. Math. Soc. 39 (2002), 145-205] . Online HTML version at http://math.ucr.edu/home/baez/octonions/node14.html.