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 ${\displaystyle {\mathfrak {g}}_{2}}$, as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups. G₂ has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14.

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.

The Lie algebra ${\displaystyle {\mathfrak {g}}_{2}}$, being the smallest exceptional simple Lie algebra, was the first of these to be discovered in the attempt to classify simple Lie algebras. On May 23, 1887, Wilhelm Killing wrote a letter to Friedrich Engel saying that he had found a 14-dimensional simple Lie algebra, which we now call ${\displaystyle {\mathfrak {g}}_{2}}$.

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