2025/10/22 by Alejandro Bravo-Doddoli, Luis C. García-Naranjo, Bravo-Doddoli, A. +3
Mathematics · #20F18 #22E25 #37J39 #53D20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2510.20006
openalex publication_date 2025/10/22 · openalex created_date 2025/10/25 · openalex updated_date 2026/08/01
Let \mathbbG be a Lie group with a normal abelian subgroup \mathbbA, and let (M,ω) be a symplectic manifold endowed with a Hamiltonian \mathbbG-action. We investigate conditions under which symplectic reduction by \mathbbG coincides with the symplectic reduction by the abelian subgroup \mathbbA. Using the reduction-by-stages framework (Marsden et al Springer Notes in Math., 1913, (2007)), we prove that, under a mild assumption, the corresponding reduced spaces are symplectomorphic if and only if they have the same dimension. Both this assumption and the dimension condition depend only on the groups \mathbbG and \mathbbA, and on the momentum value μ∈ \mathfrakg^* at which the symplectic reduction by \mathbbG is performed; in particular, they are independent of the symplectic manifold (M,ω). We then provide a broad class of examples by identifying a large family of nilpotent Lie groups, including classical Carnot groups such as the Heisenberg group and jet-space Jk(ℝn,ℝm), for which the two reduced spaces are symplectomorphic for generic momentum values.