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

Takashi Miyasaka

paper

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

arXiv:1011.0048

Abstract

A Lie group $G$ naturally acts on its Lie algebra $≫$ , called the adjoint action. In this paper, we determine the orbit types of the compact exceptional Lie group $G_{2}$ in its Lie algebra $≫_{2}$ . As results, the group $G_{2}$ has four orbit types in the Lie algebra $≫_{2}$ as $G_{2} / G_{2}, G_{2} / (U (1) \times U (1)), G_{2} / ((S p (1) \times U (1)) / Z_{2}), G_{2} / ((U (1) \times S p (1)) / Z_{2}) .$ These orbits, especially the last two orbits, are not equivalent, that is, there exists no $G_{2}$ -equivariant homeomorphism among them.

7pages,http://web.me.com/coolkai/YSTM/YSTM/YSTM.html