G2 (mathematics)

G₂ is the name of a Lie group (and also its Lie algebra). It is one of the exceptional simple Lie groups.

Table of contents

1 Roots of G2 1.1 Simple roots 2 Weyl/Coxeter group 3 Cartan matrix

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

It's Weyl/Coxeter group is the dihedral group, D₆.

Cartan matrix

See also Simple Lie group, Lie group, Weyl group, Dynkin diagram.