G2 (mathematics)

From Wikipedia, the free encyclopedia

Groups

Group theory Basic notions Subgroup Normal subgroup Quotient group Group homomorphism (semi-)direct product direct sum Discrete groups Classification of finite simple groups Cyclic group Zn Alternating group An Sporadic groups Mathieu group M11..12,M22..24 Conway group Co1..3 Janko group J1..4 Fischer group F22..24 Baby Monster group B Monster group M Continuous groups Lie group General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n) E6 E7 E8 F4 G2 SL2 Lorentz group Poincaré group Infinite dimensional groups conformal group diffeomorphism group Quantum group O(∞) SU(∞) Sp(∞)

This box: view • talk • edit

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 the holonomy group of a Riemannian metric. The manifolds of G₂ holonomy are also called G₂-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

This box: view • talk • edit