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

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

2010-10-30

arxiv.org/abs/1011.0048v1

Math.J.Okayama Univ., Vol43(2001), 17-23

Mathematics

Differential Geometry

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

Scientific paper

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.

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