Linearity of the Transverse Poisson Structure to a Coadjoint Orbit

Abstract

We prove a simple formula for the transverse Poisson structure to a coadjoint orbit (in the dual of a Lie algebra $\mathfrak{g}$) and use it in examples such as $\mathfrak{s}\mathfrak{o}\left( 4 \right)*$ and $\mathfrak{s}\mathfrak{p}\left( 4 \right)*$. We also give a sufficient condition on the isotropy subalgebra of $\mu \in \mathfrak{g}^*$ so that the transverse Poisson structureto the coadjoint orbit of μ is linear.