Coadjoint orbit

In mathematics, the coadjoint representation ${\displaystyle K}$ of a Lie group ${\displaystyle G}$ is the dual of the adjoint representation. If ${\displaystyle {\mathfrak {g}}}$ denotes the Lie algebra of ${\displaystyle G}$, the corresponding action of ${\displaystyle G}$ on ${\displaystyle {\mathfrak {g}}^{*}}$, the dual space to ${\displaystyle {\mathfrak {g}}}$, is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on ${\displaystyle G}$.

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