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Mosc. Math. J., 2004, Volume 4, Number 1, Pages 39–66 (Mi mmj142)  

This article is cited in 16 scientific papers (total in 16 papers)

Picard groups in Poisson geometry

H. Bursztyna, A. Weinsteinb

a Department of Mathematics, University of Toronto
b University of California, Berkeley

Abstract: We study isomorphism classes of symplectic dual pairs $P\leftarrow S\rightarrow\overline{P}$, where $P$ is an integrable Poisson manifold, $S$ is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simply-connected fibres. For fixed $P$, these Morita self-equivalences of $P$ form a group $Pic(P)$ under a natural “tensor product” operation. Variants of this construction are also studied, for rings (the origin of the notion of Picard group), Lie groupoids, and symplectic groupoids.

Key words and phrases: Picard group, Morita equivalence, Poisson manifold, symplectic groupoid, bimodule.

DOI: https://doi.org/10.17323/1609-4514-2004-4-1-39-66

Full text: http://www.ams.org/.../abst4-1-2004.html
References: PDF file   HTML file

Bibliographic databases:

MSC: Primary 53D17, 58H05; Secondary 16D90
Received: April 4, 2003
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Citation: H. Bursztyn, A. Weinstein, “Picard groups in Poisson geometry”, Mosc. Math. J., 4:1 (2004), 39–66

Citation in format AMSBIB
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\by H.~Bursztyn, A.~Weinstein
\paper Picard groups in Poisson geometry
\jour Mosc. Math.~J.
\yr 2004
\vol 4
\issue 1
\pages 39--66
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\crossref{https://doi.org/10.17323/1609-4514-2004-4-1-39-66}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2074983}
\zmath{https://zbmath.org/?q=an:1068.53055}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Bursztyn H., Waldmann S., “Completely positive inner products and strong Morita equivalence”, Pacific J. Math., 222:2 (2005), 201–236  crossref  mathscinet  zmath  isi
    2. Bursztyn H., Weinstein A., “Poisson geometry and Morita equivalence”, Poisson Geometry, Deformation Quantisation and Group Representations, London Mathematical Society Lecture Note Series, 323, 2005, 1  mathscinet  zmath  isi
    3. Waldmann S., “The covariant Picard groupoid in differential geometry”, Int. J. Geom. Methods Mod. Phys., 3:3 (2006), 641–654  crossref  mathscinet  zmath  isi  scopus
    4. Radko O., Shlyakhtenko D., “Picard groups of topologically stable Poisson structures”, Pacific J. Math., 224:1 (2006), 151–183  crossref  mathscinet  zmath  isi  scopus
    5. Jansen S., Waldmann S., “The $H$-covariant strong Picard groupoid”, J. Pure Appl. Algebra, 205:3 (2006), 542–598  crossref  mathscinet  zmath  isi  scopus
    6. Tseng Hsian-Hua, Zhu Chenchang, “Integrating Lie algebroids via stacks”, Compos. Math., 142:1 (2006), 251–270  crossref  mathscinet  zmath  isi  scopus
    7. Alekseev A., Meinrenken E., “Ginzburg-Weinstein via Gelfand-Zeitlin”, Journal of Differential Geometry, 76:1 (2007), 1–34  crossref  mathscinet  zmath  isi
    8. Yvette Kosmann-Schwarzbach, “Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey”, SIGMA, 4 (2008), 005, 30 pp.  mathnet  crossref  mathscinet  zmath
    9. Kosmann-Schwarzbach Y., Laurent-Gengoux C., Weinstein A., “Modular Classes of Lie Algebroid Morphisms”, Transformation Groups, 13:3–4 (2008), 727–755  crossref  mathscinet  zmath  isi  elib  scopus
    10. Sjamaar R., “Hans Duistermaat's contributions to Poisson geometry”, Bulletin of the Brazilian Mathematical Society, 42:4 (2011), 783–803  crossref  mathscinet  zmath  isi  scopus
    11. Crainic M., Struchiner I., “On the Linearization Theorem for Proper Lie Groupoids”, Ann. Sci. Ec. Norm. Super., 46:5 (2013), 723–746  crossref  mathscinet  zmath  isi
    12. Fok Ch.-K., “Picard Group of Isotropic Realizations of Twisted Poisson Manifolds”, J. Geom. Mech., 8:2 (2016), 179–197  crossref  mathscinet  zmath  isi  scopus
    13. Schwieger K., Wagner S., “Part i, Free Actions of Compact Abelian Groups on C-Algebras”, Adv. Math., 317 (2017), 224–266  crossref  zmath  isi  scopus
    14. Frejlich P., Marcut I., “On Dual Pairs in Dirac Geometry”, Math. Z., 289:1-2 (2018), 171–200  crossref  mathscinet  zmath  isi  scopus
    15. Bursztyn H., Fernandes R.L., “Picard Groups of Poisson Manifolds”, J. Differ. Geom., 109:1 (2018), 1–38  crossref  mathscinet  zmath  isi
    16. Villatoro J., “Poisson Manifolds and Their Associated Stacks”, Lett. Math. Phys., 108:3, SI (2018), 897–926  crossref  mathscinet  zmath  isi  scopus
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