The existence of a coadjoint equivariant momentum mapping for a semidirect product

Abstract:

We consider a symplectic action of a group $G$ on a symplectic manifold $P$, which admits a momentum mapping. Assume that $G$ is a semidirect product of ${G_1}$ by ${G_2}$. We prove that if the symplectic action of ${G_1}$ has a coadjoint equivariant momentum mapping, and if ${H^1}(un{k_1}:{\mathbf {R}}) = {H^2}(un{k_2}:{\mathbf {R}}) = 0$, then the symplectic action of $G$ has a coadjoint equivariant momentum mapping, where $un{k_1}$ and $un{k_2}$ are the Lie algebras of ${G_1}$ and ${G_2}$ respectively.