Abstract
Poisson manifolds are the classical analogue of associative algebras. For Poisson manifolds, symplectic realizations play a similar role as representations do for associative algebras. In this paper, the notion of Morita equivalence of Poisson manifolds, a classical analogue of Morita equivalence of algebras, is introduced and studied. It is proved that Morita equivalent Poisson manifolds have equivalent “categories” of complete symplectic realizations. For certain types of Poisson manifolds, the geometric invariants of Morita equivalence are also investigated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[B] Breen, L.: Bitoreseurs et cohomologie non abélienne. The Grothendieck Festschrift, vol.I, 401–476 (1990)
[CDW] Coste, A., Dazord, P., Weinstein, A.: Groupoïdes symplectiques. Publications du Départment de Mathématique, Université Claude Bernard LyonI (1987)
[D1] Dazord, P.: Groupoïdes symplectiques et troisième théorème de Lie “non linéaire”. Lecture Notes in Mathematics, vol.1416, pp. 39–74. Berlin, Heidelberg, New York: Springer 1990
[D2] Dazord, P.: Réalisations isotropes de Libermann. Publ. Dept. Math. Lyon (1989)
[Dix] Dixmier, J.:C *-algebras. New York: North Holland 1977
[Ka] Karasev, M. V.: Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets. Math. USSR Izv.28, 497–527 (1987)
[L] Lichnerowicz, A.: Les variétés de Poisson et leurs algèbres de Lie associées. J. Diff. Geom.12, 253–300 (1977)
[M1] Mackenzie, K.: Lie groupoids and Lie algebroids in differential geometry. LMS Lecture Notes Series, vol.124. Cambridge University Press 1987
[Mo] Morita, K.: Duality for modules and its applications to the theory of rings with minimum condition. Tokyo Kyoiku Diagaku6, 83–142 (1958)
[MiW] Mikami, K., Weinstein, A.: Moments and reduction for symplectic groupoid actions. Publ. RIMS Kyoto Univ.24, 121–140 (1988)
[MRW] Muhly, P., Renault, J., Williams, D.: Equivalence and isomorphism for groupoidC *-algebras. J. Operator Theory17, 3–22 (1987)
[Rie1] Rieffel, M. A.: Induced representations ofC *-algebras. Adv. Math.13, 176–257 (1974)
[Rie2] Rieffel, M. A.: Morita equivalence for operator algebras. Proc. Symp. Pure Math.38, 285–298 (1982)
[Rie3] Rieffel, M. A.: Applications of strong Morita equivalence to transformation groupC *-algebras. Proc. Symp. Pure Math.38, 299–310 (1982)
[Rie4] Rieffel, M. A.: Morita equivalence forC *-algebras andW *-algebras. J. Pure Appl. Algebra5, 51–96 (1974)
[W0] Weinstein, A.: The symplectic “cateogory”, Lecture Notes in Mathematics, vol.905, pp. 45–50. Berlin, Heidelberg, New York: Springer 1982
[W1] Weinstein, A.: The local structure of Poisson manifolds. J. Diff. Geom.18, 523–557 (1983)
[W2] Weinstein, A.: Symplectic groupoids and Poisson manifolds. Bull. Am. Math. Soc.16, 101–104 (1987)
[W3] Weinstein, A.: Affine Poisson structures. Int. J. Math.1, 343–360 (1990)
[W4] Weinstein, A.: Coisotropic calculus and Poisson groupoids. J. Math. Soc. Jpn.40, 705–727 (1988)
[WX] Weinstein, A., Xu, P.: Extensions of symplectic groupoids and quantization. J. Reine Angew. Math.417, 159–189 (1991)
[X1] Xu, P.: Morita equivalent symplectic groupoids, Seminaire Sud-Rhodanien à Berkeley, Symplectic geometry, groupoids, and integrable systems. Springer-MSRI publications, pp. 291–311 (1991)
[X2] Xu, P.: Symplectic realizations of Poisson manifolds. Ann. Scient. Éc. Norm. Sup. (to appear)
Author information
Authors and Affiliations
Additional information
Communicated by A. Connes
Rights and permissions
About this article
Cite this article
Xu, P. Morita equivalence of Poisson manifolds. Commun.Math. Phys. 142, 493–509 (1991). https://doi.org/10.1007/BF02099098
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02099098