2015/04/20 by Joshua Cape, Hans-Christian Herbig, Christopher Seaton · 7 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Angular momentum #Combinatorics #Diagonal #Geometry #Homogeneous #Ideal (ethics) #Mathematical analysis #Mathematical physics #Mathematics #Moment map #Physics #Pure mathematics #Quantum mechanics #Quotient #Symplectic geometry #Zero (linguistics) #math-ph #math.AC #math.AG #math.MP #math.SG #msc:13A50 #msc:20G20 #msc:37J15 #msc:53D20 #msc:57S15
paper · pdf · doi:10.3934/jgm.2016.8.13
published in The Journal of Geometric Mechanics 8(1), 13-34 (American Institute of Mathematical Sciences) · 22 pages
arxiv created 2015/04/20 · openalex publication_date 2016/02/01 · arxiv updated 2016/03/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We study the symplectic reduction of the phase space describing k particlesin ℝn with total angular momentum zero. This corresponds to the singularsymplectic quotient associated to the diagonal action of On on kcopies of T^∗ℝn at the zero value of the homogeneous quadratic moment map.We give a description of the ideal of relations of the ringof regular functions of the symplectic quotient. Using this description, wedemonstrate ℤ+-graded regular symplectomorphisms among the On- andSOn-symplectic quotients and determine which of these quotients are gradedregularly symplectomorphic to linear symplectic orbifolds. We demonstrate thatwhen n ≤ k, the zero fibre of the moment map has rational singularitiesand hence is normal and Cohen-Macaulay. We also demonstrate thatfor small values of k, the ring of regular functions on the symplecticquotient is graded Gorenstein.