Abelian instances of nonabelian symplectic reduction

TL;DR

This paper explores when the symplectic reduction of a manifold by a nonabelian group can be simplified to an abelian reduction, providing conditions and examples including nilpotent groups and Carnot groups.

Contribution

It establishes that equal dimension of reduced spaces is both necessary and sufficient for the equivalence of nonabelian and abelian symplectic reductions under certain conditions.

Findings

Dimension equality guarantees reduction equivalence.

Examples include semi-direct products and nilpotent groups.

Includes classical Carnot groups like the Heisenberg group.

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 $\mathcal{J}^k(\mathbb{R}^n,\mathbb{R}^m)$.