Adjoint orbit types of compact exceptional Lie group G2 in its Lie algebra

TL;DR

This paper classifies the types of orbits formed by the compact exceptional Lie group G2 acting on its Lie algebra, identifying four distinct orbit types and their properties.

Contribution

It explicitly determines the orbit types of G2 in its Lie algebra, including the classification of four distinct orbits and their non-equivalence.

Findings

G2 has four orbit types in its Lie algebra.

The four orbit types are explicitly described as quotient spaces.

The last two orbits are not equivalent under G2 action.

Abstract

A Lie group $G$ naturally acts on its Lie algebra $\gg$, called the adjoint action. In this paper, we determine the orbit types of the compact exceptional Lie group $G_2$ in its Lie algebra $\gg_2$. As results, the group $G_2$ has four orbit types in the Lie algebra $\gg_2$ as $$ G_2/G_2, \quad G_2/(U(1) \times U(1)), \quad G_2/((Sp(1)\times U(1))/\Z_2), \quad G_2/((U(1)\times Sp(1))/\Z_2). $$ These orbits, especially the last two orbits, are not equivalent, that is, there exists no $G_2$-equivariant homeomorphism among them.