2000/08/02 by N.P. Landsman, N. P. Landsman, Landsman, N. P. · 1 citation
Mathematics · Physics and Astronomy · #22A22 #46L08 #53D17 #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Symplectic Geometry (math.SG) #math-ph #math.MP #math.OA #math.SG #msc:22A22 #msc:46L08 #msc:53D17
paper · pdf · doi:10.48550/arxiv.math-ph/0008004
44 pages, categorical interpretation added
openalex publication_date 2000/08/02 · arxiv created 2000/11/21 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Symplectic reduction is reinterpreted as the composition of arrows in the category of integrable Poisson manifolds, whose arrows are isomorphism classes of dual pairs, with symplectic groupoids as units. Morita equivalence of Poisson manifolds amounts to isomorphism of objects in this category. This description paves the way for the quantization of the classical reduction procedure, which is based on the formal analogy between dual pairs of Poisson manifolds and Hilbert bimodules over C*-algebras, as well as with correspondences between von Neumann algebras. Further analogies are drawn with categories of groupoids (of algebraic, measured, Lie, and symplectic type). In all cases, the arrows are isomorphism classes of appropriate bimodules, and their composition may be seen as a tensor product. Hence in suitable categories reduction is simply composition of arrows, and Morita equivalence is isomorphism of objects.