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Book Review

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MathSciNet review: 1214142
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Book Information:

Author: N. V. Karasev and V. P. Maslov
Title: Nonlinear Poisson brackets, geometry and quantization
Additional book information: Translations of Math. Monographs, vol. 119, Amer. Math. Soc., Providence, RI, 1993, xi + 366 pp., ISBN 0-8218-4596-9, $170.00

References [Enhancements On Off] (What's this?)

  • [A.D] C. Albert, P. Dazord, Groupoïdes de Lie et groupoïdes symplectiques, Sém. sud-Rhodanien à Berkeley, Symplectic Geometry and Groupoids and Integrable Systems, M.S.R.I. Publications, no. 20, Springer-Verlag, Berlin and New York, 1991, pp. (1--11), MR 92m:58155.
  • [B.F.F.L.S] Bayen, Flato, Fronsdal, Lichnerowicz, Sternheimer, Deformation theory and quantization, Ann. of Physics 111 (1978), 61--110 and 111--152, MR 58:14737a,b.
  • [C] E. Cartan, Leçons sur les invariants intégraux, Hermann, Paris, 1922, 1958, 1971, MR 50:8238.
  • [Co] A. Connes, Géométrie non commutative, InterÉditions, Paris, 1990, MR 92e:58016.
  • [C.D.W] A. Coste, P. Dazord, A. Weinstein, Groupoïdes symplectiques, Public. Dept. Math. Lyon (1987), 1--62, MR 90g:58033.
  • [D.H] P. Dazord, G. Hector, Intégration symplectique des variétés de Poisson, Sém. sud-Rhodanien à Berkeley, Symplectic Geometry, Groupoids and Integrable Systems,
    M.S.R.I. Publications, no. 20, Springer-Verlag, Berlin and New York, 1991, pp. (37--72), MR 92k:58281.
  • [D.P]. P. Dazord, G. Patissier, La première classe de Chern comme obstruction à la quantification asymptotique, Sém. sud-Rhodanien à Berkeley, Symplectic Geometry and Groupoids and Integable Systems, M.S.R.I. Publications, no. 20, Springer-Verlag,
    Berlin and New York, 1991, pp. (73--97), MR 92k:58091.
  • [DW.L] M. De Wilde, P. Lecomte, Existence of star-products and of formal deformation of the Poisson-Lie algebra of arbitrary symplectic manifolds, Letters in Math. Phys. 7 (1983), 487--496, MR 85j:17021.
  • [E] C. Ehresmann, Colloque de topologie (espaces fibrés, Bruxelle, 1950), Thone, Liège, and Masson, Paris, 1951, pp. (29--55), MR 13:159.
  • [Ka] M. M. Karasev, Analogues of objects of Lie group theory for non linear Poisson brackets, Math. USSR Izvestiya, vol. 28, 1987, pp. (497--527), MR 87k:58301.
  • [Ki] A. Kirillov, Local Lie algebras, Russian Math. Surveys 31 (1976), 55--75 , MR 55:11304.
  • [Ko] B. Kostant, Quantization and unitary representations, Lecture Notes in Math., vol.
    170, Springer-Verlag, Berlin-Heidelberg-New York, 1970, pp. (87--207), MR 45:3638.
  • [Lb] P. Libermann, Problèmes d'équivalence et géométrie symplectique, III$^{\mathrm{e}}$ Rencontres du Schnepfenried, Astérisque 107/108, vol. I, 1983, pp. (43--68), MR 85k:58033.
  • [L.M] P. Libermann, C. Marle, Symplectic geometry and analytical mechanics, D. Reidel Publ. Comp., Dordrechet-Boston, 1987, MR 88c:58016.
  • [Lc] A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Diff. Geometry 12 (1977), 253--300, MR 58:18565.
  • [S] J.-M. Souriau, Structure des systèmes dynamiques, Dunod, Paris, 1970, MR 41:4866.
  • [V] I. Vaisman, Lectures on the geometry of Poisson manifolds, Progress in Math., vol. 118, Birkhäuser, Basel-Boston-Berlin, 1994, MR 95h:58057.
  • [W1] A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geometry 18 (1983), 523--557 ; MR 87k:58099, MR 86i:58059.
  • [W2] ------, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.) 16 (1987), 101--103, MR 88c:58019.
  • [W3] ------, Noncommutative geometry and geometric quantization, Symplectic Geometry and Mathematical Physics, Actes du Colloque J.-M. Souriau, Progress in Math., vol. 99, Birkhäuser, Basel-Boston-Berlin, 1991, pp. (462--476), MR 93f:58092.
  • [W.X] A. Weinstein, P. Xu, Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), 159--189, MR 92k:58094.

Review Information:

Reviewer: P. Libermann
Affiliation: University Paris VII
Journal: Bull. Amer. Math. Soc. 33 (1996), 101-105
DOI: https://doi.org/10.1090/S0273-0979-96-00619-2
Review copyright: © Copyright 1996 American Mathematical Society
American Mathematical Society