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Quantization of Singular Symplectic Quotients pp 137–180Cite as

Birkhäuser

Quantized reduction as a tensor product

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Part of the book series: Progress in Mathematics ((PM,volume 198))

Abstract

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 overC*-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.

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Authors and Affiliations

  1. Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, NL-1018 TV, Amsterdam, The Netherlands

    N. P. Landsman

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  1. N. P. Landsman
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Editors and Affiliations

  1. Korteweg-de Vries Instituut voor Wiskunde, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV, Amsterdam, The Netherlands

    N. P. Landsman

  2. Mathematisch-Naturwissenschaftliche Fakultät II Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    M. Pflaum

  3. Fakultät für Mathematik und Informatik, Universität Mannheim, D7, 27, 68131, Mannheim, Germany

    M. Schlichenmaier

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Landsman, N.P. (2001). Quantized reduction as a tensor product. In: Landsman, N.P., Pflaum, M., Schlichenmaier, M. (eds) Quantization of Singular Symplectic Quotients. Progress in Mathematics, vol 198. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8364-1_7

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  • DOI: https://doi.org/10.1007/978-3-0348-8364-1_7

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9535-4

  • Online ISBN: 978-3-0348-8364-1

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