G₂

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

:egin{pmatrix}2&-3\-1&2end{pmatrix}

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.


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