G₂ (mathematics)

G₂ (mathematics)

In mathematics, G₂ is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras, as well as some algebraic groups. 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.

In older books and papers, G₂ is sometimes denoted by E₂.