G2 (mathematics)

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In mathematics, G₂ is the name of some Lie groups and also 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.

Contents 1 Real forms 2 Algebra 2.1 Dynkin diagram 2.2 Roots of G2 2.3 Weyl/Coxeter group 2.4 Cartan matrix 2.5 Special holonomy 3 References

[edit] 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.

[edit] Algebra

[edit] Dynkin diagram

[edit] 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)

[edit] Weyl/Coxeter group

Its Weyl/Coxeter group is the dihedral group, D₁₂ of order 12.

[edit] Cartan matrix

[edit] 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.

[edit] 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 | E8 | F4 | G2

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