Abelian instances of nonabelian symplectic reduction

Mathematics > Symplectic Geometry [Submitted on 22 Oct 2025 (v1), last revised 16 Jun 2026 (this version, v2)] Title:Abelian instances of nonabelian symplectic reduction View PDF HTML (experimental)Abstract:Let $\mathbb{G}$ be a Lie group with a normal abelian subgroup $\mathbb{A}$, and let $(M,\omega)$ be a symplectic manifold endowed with a Hamiltonian $\mathbb{G}$-action. We investigate conditions under which symplectic reduction by $\mathbb{G}$ coincides with the symplectic reduction by the abelian subgroup $\mathbb{A}$. 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 $\mathbb{G}$ and $\mathbb{A}$, and on the momentum value $\mu\in \mathfrak{g}^*$ at which the symplectic reduction by $\mathbb{G}$ is performed; in particular, they are independent of the symplectic manifold $(M,\omega)$. 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 $\mathcal{J}^k(\mathbb{R}^n,\mathbb{R}^m)$, for which the two reduced spaces are symplectomorphic for generic momentum values. Submission history From: Luis García-Naranjo [view email][v1] Wed, 22 Oct 2025 20:11:46 UTC (36 KB) [v2] Tue, 16 Jun 2026 14:16:40 UTC (40 KB) Current browse context: math.SG References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.