Abelian instances of nonabelian symplectic reduction
Abstract
Consider a Lie group $\mathbb{G}$ with a normal abelian subgroup $\mathbb{A}$. Suppose that $\mathbb{G}$ acts on a Hamiltonian fashion on a symplectic manifold $(M,\omega)$. Such action can be restricted to a Hamiltonian action of $\mathbb{A}$ on $M$. This work investigates the conditions under which the (generally nonabelian) symplectic reduction of $M$ by $\mathbb{G}$ is equivalent to the (abelian) symplectic reduction of $M$ by $\mathbb{A}$. While the requirement that the symplectically reduced spaces share the same dimension is evidently necessary, we prove that it is, in fact, sufficient. We then provide classess of examples where such equivalence holds for generic momentum values. These examples include certain semi-direct products and a large family of nilpotent groups which includes some classical Carnot groups, like the Heisenberg group and the jet space $\mathcal{J}^k(\mathbb{R}^n,\mathbb{R}^m)$.