Questions tagged [coadjoint-action]

The coadjoint orbit of the Lie group $SU(n)$ is given by $\mathcal{O}_{SU(n)} = SU(n)/\mathbb{T}_{n-1} \cong \mathbb{C}\mathbb{P}^{n-1} \ltimes \mathcal{O}_{SU(n-1)}$. It is well known that this orbit $\mathcal{O}$ is a homogeneous Kähler manifold,…

Let $\mathfrak {g}$ be a Lie bialgebra. Then $\mathfrak {g}^{\ast}$ is also a Lie bialgebra which is dual to $\mathfrak {g}.$ Let the brackets on $\mathfrak {g}$ and $\mathfrak {g}^{\ast}$ be denoted by $b$ and $b'$ respectively. Then how to define…

Let $G\times \mathfrak{g}^*\to \mathfrak{g}^*$ be the coadjoint action defined by $f\cdot q=Ad^*_fq$ where $Ad:G\to GL(\mathfrak{g})$ is the adjoint representation of $G$ with dual $Ad^*$. For $p\in \mathfrak{g}^*$, let $W_p$ be the coadjoint orbit…

The natural action of $SU(n+1)$ is transitive on $\mathbb{CP}^n$. This suggests that the latter can be realized as a coadjoint orbit of the former. I am trying make this explicit. I have been able to show that $\mathbb{CP}^n\cong…