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Classification of invariant star products up to equivariant Morita equivalence on symplectic manifolds. (English) Zbl 1251.53057
This paper gives the full classification of invariant star products up to equivariant Morita equivalence on a symplectic manifold. After some recalls about Morita theory and groupoid morphisms, the authors reformulate the criteria on the existence of quantum momentum maps in terms of equivariant cohomology. They then compute explicitly the equivariant Picard groupoid in terms of the Picard groupoid. They study in details the case of star products on symplectic manifolds. The main result of this paper enables the authors to classify the star-products up to Morita equivalence.

MSC:
53D55 Deformation quantization, star products
16D90 Module categories in associative algebras
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[1] Ara P.: Morita equivalence for rings with involution. Alg. Represent. Theory 2, 227–247 (1999) · Zbl 0944.16005 · doi:10.1023/A:1009958527372
[2] Bass H.: Algebraic K-Theory. W. A. Benjamin Inc., New York (1968)
[3] Bayen F., Flato M., Frønsdal C., Lichnerowicz A., Sternheimer D.: Deformation theory and quantization. Ann. Phys. 111, 61–151 (1978) · Zbl 0377.53024 · doi:10.1016/0003-4916(78)90224-5
[4] Bertelson M., Bieliavsky P., Gutt S.: Parametrizing equivalence classes of invariant star products. Lett. Math. Phys. 46, 339–345 (1998) · Zbl 0943.53051 · doi:10.1023/A:1007598606137
[5] Bursztyn H.: Semiclassical geometry of quantum line bundles and Morita equivalence of star products. Int. Math. Res. Not. 16, 821–846 (2002) · Zbl 1031.53120 · doi:10.1155/S1073792802108014
[6] Bursztyn, H., Dolgushev, V., Waldmann, S.: Morita equivalence and characteristic classes of star products. Preprint (2009). arXiv:0909:4259 · Zbl 1237.53080
[7] Bursztyn H., Waldmann S.: The characteristic classes of Morita equivalent star products on symplectic manifolds. Commun. Math. Phys. 228, 103–121 (2002) · Zbl 1036.53068 · doi:10.1007/s002200200657
[8] Bursztyn H., Waldmann S.: Bimodule deformations, Picard groups and contravariant connections. K-Theory 31, 1–37 (2004) · Zbl 1054.53101 · doi:10.1023/B:KTHE.0000021354.07931.64
[9] Bursztyn H., Waldmann S.: Completely positive inner products and strong Morita equivalence. Pac. J. Math. 222, 201–236 (2005) · Zbl 1111.53071 · doi:10.2140/pjm.2005.222.201
[10] Dolgushev V.A.: Covariant and equivariant formality theorems. Adv. Math. 191, 147–177 (2005) · Zbl 1116.53065 · doi:10.1016/j.aim.2004.02.001
[11] Fedosov B.V.: A simple geometrical construction of deformation quantization. J. Differ. Geom. 40, 213–238 (1994) · Zbl 0812.53034
[12] Guillemin V.W., Sternberg S.: Supersymmetry and Equivariant de Rham Theory. Springer, Berlin (1999) · Zbl 0934.55007
[13] Gutt S., Rawnsley J.: Equivalence of star products on a symplectic manifold: an introduction to Deligne’s Čech cohomology classes. J. Geom. Phys. 29, 347–392 (1999) · Zbl 1024.53057 · doi:10.1016/S0393-0440(98)00045-X
[14] Gutt S., Rawnsley J.: Natural star products on symplectic manifolds and quantum moment maps. Lett. Math. Phys. 66, 123–139 (2003) · Zbl 1064.53061 · doi:10.1023/B:MATH.0000017717.51035.f1
[15] Jansen, S.: H-Äquivariante Morita-Äquivalenz und Deformationsquantisierung. PhD thesis, Fakultät für Mathematik und Physik, Physikalisches Institut, Albert-Ludwigs-Universität, Freiburg (2006)
[16] Jansen S., Waldmann S.: The H-covariant strong Picard groupoid. J. Pure Appl. Algebra 205, 542–598 (2006) · Zbl 1108.53056 · doi:10.1016/j.jpaa.2005.07.015
[17] Jurčo B., Schupp P., Wess J.: Noncommutative line bundles and Morita equivalence. Lett. Math. Phys. 61, 171–186 (2002) · Zbl 1036.53070 · doi:10.1023/A:1021244731214
[18] Kostant B.: Quantization and unitary representation. Part I: prequantization. In: Taam, C.T. (ed) Lectures in Modern Analysis and Application. Lecture Notes in Mathematics, vol. 170, pp. 87–208. Springer, Berlin (1970)
[19] Lashof R.: Equivariant prequantization. In: Dazord, P., Weinstein, A. (eds) Symplectic Geometry, Groupoids, and Integrable Systems. Mathematical Sciences Research Institute Publications, vol. 20, pp. 193–207. Springer, New York (1991) · Zbl 0731.58030
[20] Müller-Bahns M.F., Neumaier N.: Some remarks on $${\(\backslash\)mathfrak{g}}$$ -invariant Fedosov star products and quantum momentum mappings. J. Geom. Phys. 50, 257–272 (2004) · Zbl 1078.53100 · doi:10.1016/j.geomphys.2003.10.003
[21] Waldmann S.: Morita equivalence of Fedosov star products and deformed Hermitian vector bundles. Lett. Math. Phys. 60, 157–170 (2002) · Zbl 1009.53063 · doi:10.1023/A:1016109723843
[22] Waldmann S.: The covariant Picard groupoid in differential geometry. Int. J. Geom. Methods Mod. Phys. 3(3), 641–654 (2006) · Zbl 1135.53061 · doi:10.1142/S0219887806001314
[23] Waldmann S.: Poisson-Geometrie und Deformationsquantisierung. Eine Einführung. Springer, Heidelberg (2007) · Zbl 1139.53001
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