Dippell, Marvin; Esposito, Chiara; Waldmann, Stefan

Coisotropic Triples, Reduction and Classical Limit

Doc. Math. 24, 1811-1853 (2019)
DOI: 10.25537/dm.2019v24.1811-1853
Communicated by Eckhard Meinrenken


Coisotropic reduction from Poisson geometry and deformation quantization is cast into a general and unifying algebraic framework: we introduce the notion of coisotropic triples of algebras for which a reduction can be defined. This allows to construct also a notion of bimodules for such triples leading to bicategories of bimodules for which we have a reduction functor as well. Morita equivalence of coisotropic triples of algebras is defined as isomorphism in the ambient bicategory and characterized explicitly. Finally, we investigate the classical limit of coisotropic triples of algebras and their bimodules and show that classical limit commutes with reduction in the bicategory sense.

Mathematics Subject Classification

53D55, 53D20, 16D90


coisotropic, reduction, quantization, Morita


  • 1. Bayen, F., Flato, M., Frønsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. Ann. Phys. 111 (1978), 61-151. DOI 10.1016/0003-4916(78)90225-7; zbl 0377.53025; MR0496158.
  • 2. Bénabou, J.: Introduction to Bicategories. In: Reports of the Midwest Category Seminar, 1-77. Springer-Verlag, 1967. DOI 10.1007/BFb0074299; zbl 1375.18001; MR0220789.
  • 3. Bénabou, J., Davis, R., Dold, A., Isbell, J., MacLane, S., Oberst, U., Roos, J.-E. (eds.): Reports of the Midwest Category Seminar, vol. 47, in Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1967. DOI 10.1007/BFb0074298; zbl 0165.33001.
  • 4. Bordemann, M.: (Bi)Modules, morphismes et réduction des star-produits: le cas symplectique, feuilletages et obstructions. Preprint (2004), 135 pages. arxiv math/0403334.
  • 5. Bordemann, M.: (Bi)Modules, morphisms, and reduction of star-products: the symplectic case, foliations, and obstructions. Trav. Math. 16 (2005), 9-40. zbl 1099.53061; MR2223149.
  • 6. Bordemann, M., Herbig, H.-C., Pflaum, M. J.: A homological approach to singular reduction in deformation quantization. In: Chéniot, D., Dutertre, N., Murolo, C., Trotman, D., Pichon, A. (eds.): Singularity theory, 443-461. World Scientific Publishing, Hackensack, 2007. Proceedings of the Singularity School and Conference held in Marseille, January 24-February 25, 2005. Dedicated to Jean-Paul Brasselet on his 60th birthday. zbl 1128.53060; MR2342922; arxiv math-ph/0603078.
  • 7. Bordemann, M., Herbig, H.-C., Waldmann, S.: BRST cohomology and phase space reduction in deformation quantization. Commun. Math. Phys. 210 (2000), 107-144. DOI 10.1007/s002200050774; zbl 0961.53046; MR1748172; arxiv math/9901015.
  • 8. Bursztyn, H.: Semiclassical geometry of quantum line bundles and Morita equivalence of star products. Int. Math. Res. Not. no. 16 (2002), 821-846. DOI 10.1155/S1073792802108014; zbl 1031.53120; MR1891209; arxiv math/0105001.
  • 9. Bursztyn, H., Fernandes, R. L.: Picard groups of Poisson manifolds. J. Diff. Geom. 109.1 (2018), 1-38. DOI 10.4310/jdg/1525399215; zbl 1392.53088; MR3798714; arxiv 1509.03780.
  • 10. Bursztyn, H., Waldmann, S.: Algebraic Rieffel induction, formal Morita equivalence and applications to deformation quantization. J. Geom. Phys. 37 (2001), 307-364. DOI 10.1016/S0393-0440(00)00035-8; zbl 1039.46052; MR1811148; arxiv math/9912182.
  • 11. Bursztyn, H., Waldmann, S.: The characteristic classes of Morita equivalent star products on symplectic manifolds. Commun. Math. Phys. 228 (2002), 103-121. DOI 10.1007/s002200200657; zbl 1036.53068; MR1911250; arxiv math/0106178.
  • 12. Bursztyn, H., Waldmann, S.: Bimodule deformations, Picard groups and contravariant connections. K-Theory 31 (2004), 1-37. DOI 10.1023/B:KTHE.0000021354.07931.64; zbl 1054.53101; MR2050877; arxiv math/0207255.
  • 13. Bursztyn, H., Waldmann, S.: Completely positive inner products and strong Morita equivalence. Pacific J. Math. 222 (2005), 201-236. DOI 10.2140/pjm.2005.222.201; zbl 1111.53071; MR2225070; arxiv math/0309402.
  • 14. Bursztyn, H., Weinstein, A.: Picard groups in Poisson geometry. Moscow Math. J. 4 (2004), 39-66. zbl 1068.53055; MR2074983; arxiv math/0304048.
  • 15. Burzstyn, H., Dolgushev, V., Waldmann, S.: Morita equivalence and characteristic classes of star products. J. reine angew. Math. 662 (2012), 95-163. DOI 10.1515/CRELLE.2011.089; zbl1237.53080; MR2876262; arxiv 0909.4259.
  • 16. Cattaneo, A. S.: On the integration of Poisson manifolds, Lie algebroids, and coisotropic submanifolds. Lett. Math. Phys. 67 (2004), 33-48. DOI 10.1023/B:MATH.0000027690.76935.f3; zbl 1059.53064; MR2063018; arxiv math/0308180.
  • 17. Cattaneo, A. S., Felder, G.: Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model. Lett. Math. Phys. 69 (2004), 157-175. DOI 10.1007/s11005-004-0609-7; zbl 1065.53063; MR2104442; arxiv math/0309180.
  • 18. Cattaneo, A. S., Felder, G.: Relative formality theorem and quantisation of coisotropic submanifolds. Adv. Math. 208 (2007), 521-548. DOI 10.1016/j.aim.2006.03.010; zbl 1106.53060; MR2304327; arxiv math/0501540.
  • 19. Ciccoli, N.: Quantization of coisotropic subgroups. Lett. Math. Phys. 42 (1997), 123-138. DOI 10.1023/A:1007352218739; zbl 0951.17009; MR1479356.
  • 20. Ciccoli, N.: Some comments on coisotropic triples. Private communication, 2019.
  • 21. Ciccoli, N., Gavarini, F.: Quantum duality principle for coisotropic subgroups and Poisson quotients. Adv. Math. 199 (2006), 104-135. DOI 10.1016/j.aim.2005.01.009; zbl 1137.58003; MR2187400; arxiv math/0412465.
  • 22. Dirac, P. A.M.: Generalized Hamiltonian dynamics. Canad. J. Math. 2 (1950), 129-148. DOI 10.4153/CJM-1950-012-1; zbl 0036.14104; MR0043724.
  • 23. Dirac, P. A. M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, Yeshiva University, New York, 1964.
  • 24. Esposito, C.: Poisson reduction. In: Kielanowski, P., Bieliavsky, P., Odesskii, A., Schlichenmaier, M., Voronov, T. (eds.): Geometric methods in physics, Trends in Mathematics, 131-142. Birkhäuser/Springer, Cham, 2014. Selected papers from the XXXII Workshop (WGMP) held in Bialowieza, June 30-July 6, 2013. DOI 10.1007/978-3-319-06248-8_11; zbl 1337.53099; MR3587686; arxiv 1106.3878.
  • 25. Esposito, C., de Kleijn, N., Schnitzer, J.: A proof of Tsygan's formality conjecture for Hamiltonian actions. Preprint (2018), 9 pages. arxiv 1812.00403.
  • 26. Fedosov, B. V.: Non-abelian reduction in deformation quantization. Lett. Math. Phys. 43 (1998), 137-154. DOI 10.1023/A:1007451214380; zbl 0964.53055; MR1607363.
  • 27. Gruski, N.: An algebraic theory of tricategories. PhD thesis, University of Chicago, Chicago, 2007. MR2717302.
  • 28. Gutt, S., Rawnsley, J.: Natural star products on symplectic manifolds and quantum moment maps. Lett. Math. Phys. 66 (2003), 123-139. DOI 10.1023/B:MATH.0000017717.51035.f1; zbl 1064.53061; MR2064595; arxiv math/0304498.
  • 29. Gutt, S., Waldmann, S.: Involutions and representations for reduced quantum algebras. Adv. Math. 224 (2010), 2583-2644. DOI 10.1016/j.aim.2010.02.009; zbl 1236.53070; MR2652217; arxiv 0911.1649.
  • 30. Jansen, S., Neumaier, N., Schaumann, G., Waldmann, S.: Classification of invariant star products up to equivariant Morita equivalence on symplectic manifolds. Lett. Math. Phys. 100 (2012), 203-236. DOI 10.1007/s11005-011-0536-3; zbl 1251.53057; MR2912481; arxiv 1004.0875.
  • 31. Jansen, S., Waldmann, S.: The \(H\)-covariant strong Picard groupoid. J. Pure Appl. Alg. 205 (2006), 542-598. DOI 10.1016/j.jpaa.2005.07.015; zbl 1108.53056; MR2210219; arxiv math/0409130.
  • 32. Leinster, T.: Higher Operads, Higher Categories, vol. 298 in London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2004. zbl 1160.18001; MR2094071; arxiv math/0305049.
  • 33. Lu, J.-H.: Multiplicative and affine Poisson structures on Lie groups. PhD thesis, University of California, Berkeley, 1990. MR2685337.
  • 34. Lu, J.-H.: Moment maps at the quantum level. Comm. Math. Phys. 157 (1993), 389-404. DOI 10.1007/BF02099767, zbl 0801.17019, MR1244874.
  • 35. Marsden, J. E., Ratiu, T. S.: Introduction to Mechanics and Symmetry. Texts in applied mathematics, no. 17. Springer-Verlag, New York, Heidelberg, 1999. zbl 0933.70003; MR1723696.
  • 36. Müller-Bahns, M. F., Neumaier, N.: Some remarks on \(\mathfrak{g} \)-invariant Fedosov star products and quantum momentum mappings. J. Geom. Phys. 50 (2004), 257-272. DOI 10.1016/j.geomphys.2003.10.003; zbl 1078.53100; MR2078228; arxiv math/0301101.
  • 37. Ortega, J.-P., Ratiu, T. S.: Momentum Maps and Hamiltonian Reduction, vol. 222 in Progress in Mathematics. Birkhäuser, Boston, 2004. zbl 1241.53069; MR2021152.
  • 38. Reichert, T.: Characteristic classes of star products on Marsden-Weinstein reduced symplectic manifolds. Lett. Math. Phys. 107 (2017), 643-658. DOI 10.1007/s11005-016-0921-z; zbl 1365.53081; MR3623275; arxiv 1605.04147.
  • 39. Reichert, T., Waldmann, S.: Classification of equivariant star products on symplectic manifolds. Lett. Math. Phys. 106 (2016), 675-692. DOI 10.1007/s11005-016-0834-x; zbl 1341.53123; MR3490952; arxiv 1507.00146.
  • 40. Waldmann, S.: Poisson-Geometrie und Deformationsquantisierung. Eine Einführung. Springer-Verlag, Heidelberg, Berlin, New York, 2007. DOI 10.1007/978-3-540-72518-3; zbl 1139.53001.
  • 41. Xu, P.: Morita equivalence of Poisson manifolds. Commun. Math. Phys. 142 (1991), 493-509. DOI 10.1007/BF02099098; zbl 0746.58034; MR1138048.


Dippell, Marvin
Julius Maximilian University of Würzburg, Department of Mathematics, Chair of Mathematics X (Mathematical Physics), Emil-Fischer-Straße 31, 97074 Würzburg, Germany
Esposito, Chiara
Dipartimento di Matematica, Università degli Studi di Salerno, via Giovanni Paolo II, 123 84084 Fisciano (SA), Italy
Waldmann, Stefan
Julius Maximilian University of Würzburg, Department of Mathematics, Chair of Mathematics X (Mathematical Physics), Emil-Fischer-Straße 31, 97074 Würzburg, Germany