2002/04/11 by Juan‐Pablo Ortega, Juan-Pablo Ortega, Ortega, Juan-Pablo
Mathematics · #37J15 #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) #math.DG #math.SG #msc:37J15 #msc:53D20
paper · pdf · doi:10.48550/arxiv.math/0204154
8 pages. To appear in C. R. Acad. Sci. Paris Sér. I Math
arxiv created 2002/04/11 · openalex publication_date 2002/04/11 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
During the last thirty years, symplectic or Marsden--Weinstein reduction has been a major tool in the construction of new symplectic manifolds and in the study of mechanical systems with symmetry. This procedure has been traditionally associated to the canonical action of a Lie group on a symplectic manifold, in the presence of a momentum map. In this note we show that the symplectic reduction phenomenon has much deeper roots. More specifically, we will find symplectically reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. On other words, the right category to obtain symplectically reduced spaces is that of Poisson manifolds acted canonically upon by a Lie group.