Given a Lie group $G$, it acts smoothly on the dual ${g}^{*}$ of its Lie algebra $g$ by the coadjoint action. The orbits of that action are called coadjoint orbits. They are especially important in the orbit method of representation theory or, more generally, geometric quantization.
The dual ${g}^{*}$ of a (say finite-dimensional real) Lie algebra has a structure of a Poisson manifold with the Poisson structure due to A. Kirillov and Souriau, namely for any $a\in {g}^{*}$,
$\left\{f,g\right\}\left(a\right):=⟨\left[d{f}_{a},d{g}_{a}\right],a⟩$\{ f, g\}(a) := \langle [d f_a, d g_a],a\rangle