Morita equivalence of Poisson manifolds. (English) Zbl 0746.58034

Xu, Ping

Commun. Math. Phys. 142, No. 3, 493-509 (1991).

The author uses the analogy between Poisson manifolds and their symplectic realizations on the one side and associative algebras (in particular, \(C^*\)-algebras) and their representations on the other side. The notion of Morita equivalence of Poisson manifolds is introduced by analogy of Morita equivalence of algebras. It is proved that Morita equivalent Poisson manifolds have equivalent “categories” of complete symplectic realizations. The geometric invariants of Morita equivalence are also investigated; namely it is shown that for a Poisson manifold in which the characteristic foliations are trivial fibrations the variation lattice of symplectic structures along symplectic leaves is a complete Morita equivalence invariant.

Reviewer: V.B.Marenich (Novosibirsk)

MSC:

37J99	Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
53D50	Geometric quantization
16D90	Module categories in associative algebras

Keywords:

Poisson manifolds; symplectic realizations; Morita equivalence

References:

[1]	[B] Breen, L.: Bitoreseurs et cohomologie non abélienne. The Grothendieck Festschrift, vol.I, 401–476 (1990)
[2]	[CDW] Coste, A., Dazord, P., Weinstein, A.: Groupoïdes symplectiques. Publications du Départment de Mathématique, Université Claude Bernard LyonI (1987)
[3]	[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
[4]	[D2] Dazord, P.: Réalisations isotropes de Libermann. Publ. Dept. Math. Lyon (1989)
[5]	[Dix] Dixmier, J.:C *-algebras. New York: North Holland 1977 · Zbl 0372.46058
[6]	[Ka] Karasev, M. V.: Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets. Math. USSR Izv.28, 497–527 (1987) · Zbl 0624.58007
[7]	[L] Lichnerowicz, A.: Les variétés de Poisson et leurs algèbres de Lie associées. J. Diff. Geom.12, 253–300 (1977) · Zbl 0405.53024
[8]	[M1] Mackenzie, K.: Lie groupoids and Lie algebroids in differential geometry. LMS Lecture Notes Series, vol.124. Cambridge University Press 1987 · Zbl 0683.53029
[9]	[Mo] Morita, K.: Duality for modules and its applications to the theory of rings with minimum condition. Tokyo Kyoiku Diagaku6, 83–142 (1958) · Zbl 0080.25702
[10]	[MiW] Mikami, K., Weinstein, A.: Moments and reduction for symplectic groupoid actions. Publ. RIMS Kyoto Univ.24, 121–140 (1988) · Zbl 0659.58016
[11]	[MRW] Muhly, P., Renault, J., Williams, D.: Equivalence and isomorphism for groupoidC *-algebras. J. Operator Theory17, 3–22 (1987) · Zbl 0645.46040
[12]	[Rie1] Rieffel, M. A.: Induced representations ofC *-algebras. Adv. Math.13, 176–257 (1974) · Zbl 0284.46040
[13]	[Rie2] Rieffel, M. A.: Morita equivalence for operator algebras. Proc. Symp. Pure Math.38, 285–298 (1982)
[14]	[Rie3] Rieffel, M. A.: Applications of strong Morita equivalence to transformation groupC *-algebras. Proc. Symp. Pure Math.38, 299–310 (1982)
[15]	[Rie4] Rieffel, M. A.: Morita equivalence forC *-algebras andW *-algebras. J. Pure Appl. Algebra5, 51–96 (1974) · Zbl 0295.46099
[16]	[W0] Weinstein, A.: The symplectic ”cateogory”, Lecture Notes in Mathematics, vol.905, pp. 45–50. Berlin, Heidelberg, New York: Springer 1982
[17]	[W1] Weinstein, A.: The local structure of Poisson manifolds. J. Diff. Geom.18, 523–557 (1983) · Zbl 0524.58011
[18]	[W2] Weinstein, A.: Symplectic groupoids and Poisson manifolds. Bull. Am. Math. Soc.16, 101–104 (1987) · Zbl 0618.58020
[19]	[W3] Weinstein, A.: Affine Poisson structures. Int. J. Math.1, 343–360 (1990) · Zbl 0725.58014
[20]	[W4] Weinstein, A.: Coisotropic calculus and Poisson groupoids. J. Math. Soc. Jpn.40, 705–727 (1988) · Zbl 0642.58025
[21]	[WX] Weinstein, A., Xu, P.: Extensions of symplectic groupoids and quantization. J. Reine Angew. Math.417, 159–189 (1991) · Zbl 0722.58021
[22]	[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)
[23]	[X2] Xu, P.: Symplectic realizations of Poisson manifolds. Ann. Scient. Éc. Norm. Sup. (to appear)