G2 (mathematics)

From Free net encyclopedia

In mathematics, G₂ is the name of a Lie group and also its Lie algebra <math>\mathfrak{g}_2</math>. It is the smallest of the five exceptional simple Lie groups. G₂ has rank 2 and dimension 14. Its center is the trivial subgroup. Its outer automorphism group is the trivial group. Its fundamental representation is 7-dimensional.

G₂ can be described as the automorphism group of the octonion algebra or, equivalently, as the subgroup of <math>SO(7)</math> that preserves any chosen particular vector in its 8-dimensional real spinor representation.

Contents 1 Algebra 1.1 Dynkin diagram 1.2 Roots of G2 1.3 Weyl/Coxeter group 1.4 Cartan matrix 1.5 Special holonomy 2 References

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.

Cartan matrix

<math>

\begin{pmatrix} 2&-3\\ -1&2 \end{pmatrix} </math>

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.

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