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arXiv · 1011.0048

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

Takashi Miyasaka
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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.

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Takashi Miyasaka. 2010-10-30. Adjoint orbit types of compact exceptional Lie group G2 in its Lie algebra. https://arxiv.org/abs/1011.0048

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