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Book Review

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Book Information

Author(s): 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, $170.00, 0-8218-4596-9

References:

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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]
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[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.

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A. Kirillov, Local Lie algebras, Russian Math. Surveys 31 (1976), 55--75 , MR 55:11304.

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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.

Additional Information:

Reviewer(s):
P. Libermann
Affiliation: University Paris VII

Review Information:
Journal: Bull. Amer. Math. Soc. 33 (1996), 101-105.
DOI: 10.1090/S0273-0979-96-00619-2
PII: S 0273-0979(96)00619-2
Copyright of article: Copyright 1996, American Mathematical Society

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