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Abelian instances of nonabelian symplectic reduction

A. Bravo-Doddoli, L. C. García-Naranjo, E. Rigato

TL;DR

The paper analyzes when the symplectic reduction of a symplectic manifold by a nonabelian group $\mathbb{G}$ coincides with the abelian reduction by a normal abelian subgroup $\mathbb{A}$. It proves that, for free, proper Hamiltonian actions with connected stabilizers, the two reductions are symplectomorphic whenever their dimensions match, a condition shown to be sufficient via Reduction by Stages. The work then identifies broad classes of groups where generic momentum values satisfy the dimension condition, including semidirect products and $\,\mathbb{A}$-simple metabelian nilpotent (Carnot-type) groups such as the Heisenberg group and jet spaces $\mathcal{J}^k(\mathbb{R}^n,\mathbb{R}^m)$. It provides explicit theorems and constructions clarifying when abelian reduction suffices and when coadjoint orbits correspond to standard cotangent bundles with magnetic terms. The results simplify reductions in practical problems (e.g., planar vortex dynamics, sub-Riemannian geodesics) and illuminate the geometric structure of generic coadjoint orbits for a wide family of groups.

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

Abelian instances of nonabelian symplectic reduction

TL;DR

The paper analyzes when the symplectic reduction of a symplectic manifold by a nonabelian group coincides with the abelian reduction by a normal abelian subgroup . It proves that, for free, proper Hamiltonian actions with connected stabilizers, the two reductions are symplectomorphic whenever their dimensions match, a condition shown to be sufficient via Reduction by Stages. The work then identifies broad classes of groups where generic momentum values satisfy the dimension condition, including semidirect products and -simple metabelian nilpotent (Carnot-type) groups such as the Heisenberg group and jet spaces . It provides explicit theorems and constructions clarifying when abelian reduction suffices and when coadjoint orbits correspond to standard cotangent bundles with magnetic terms. The results simplify reductions in practical problems (e.g., planar vortex dynamics, sub-Riemannian geodesics) and illuminate the geometric structure of generic coadjoint orbits for a wide family of groups.

Abstract

Consider a Lie group with a normal abelian subgroup . Suppose that acts on a Hamiltonian fashion on a symplectic manifold . Such action can be restricted to a Hamiltonian action of on . This work investigates the conditions under which the (generally nonabelian) symplectic reduction of by is equivalent to the (abelian) symplectic reduction of by . 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 .
Paper Structure (25 sections, 15 theorems, 120 equations, 1 table)

This paper contains 25 sections, 15 theorems, 120 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Usefulness of the construction.
  4. Relation with Reduction by Stages of Marsden, Misioł ek, Perlmutter, Ratiu Marsden1998stages.
  5. Structure of the paper
  6. Equivalent symplectic reductions
  7. Preliminaries.
  8. A useful lemma
  9. $\mathfrak{a}$ as a maximal isotropic subspace.
  10. Definition and properties of $\mathop{\mathrm{T}}\nolimits_{\mu}$.
  11. Proof of Lemma \ref{['lemm:SSH']}.
  12. Proof of Theorem \ref{['th:main-new']}
  13. Alternative formulation of the Theorem
  14. Calculation of coadjoint orbits
  15. Examples I: Semidirect products
  16. ...and 10 more sections

Key Result

Theorem A

Let $\mathbb{G}$ be a Lie group with an abelian, normal, and regular subgroup $\mathbb{A}$. Suppose $\mathbb{G}$ defines a free and proper Hamiltonian action on the symplectic manifold $(M,\omega)$ with equivariant momentum map $J_\mathbb{G}:M\to \mathfrak{g}^*$. Let $\mu\in J_\mathbb{G}(M)\subset \

Theorems & Definitions (28)

  • Theorem A
  • Remark 1
  • Lemma 1
  • Remark 2
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 3
  • Proposition 4
  • ...and 18 more