Book Review
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Book Information:
Author:
Izu Vaisman
Title:
Lectures on the geometry of Poisson manifolds
Additional book information:
Progress in Mathematics, vol. 118,
Birkhäuser, Basel and Boston,
1994,
vi + 205 pp.,
ISBN 3-7643-5016-4,
$59.00$
Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 1–3, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1975 edition. MR 979493
Jean-Luc Brylinski, A differential complex for Poisson manifolds, J. Differential Geom. 28 (1988), no. 1, 93–114. MR 950556
Jack F. Conn, Normal forms for smooth Poisson structures, Ann. of Math. (2) 121 (1985), no. 3, 565–593. MR 794374, DOI 10.2307/1971210
[D] P. M. Dirac, Lectures on quantum mechanics, Befer Graduate School Sci. Yeshiva Univ., New York, 1964.
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
Robert Hermann, Cartan connections and the equivalence problem for geometric structures, Contributions to Differential Equations 3 (1964), 199–248. MR 165459
A. A. Kirillov, Local Lie algebras, Uspehi Mat. Nauk 31 (1976), no. 4(190), 57–76 (Russian). MR 0438390
[Kar] M. V. Karasev, Analogues of the objects of Lie group theory for nonlinear Poisson brackets, Math. USSR-Izv. 28 (1987), 497--527.
M. V. Karasëv and V. P. Maslov, Nonlinear Poisson brackets, Translations of Mathematical Monographs, vol. 119, American Mathematical Society, Providence, RI, 1993. Geometry and quantization; Translated from the Russian by A. Sossinsky [A. B. Sosinskiĭ] and M. Shishkova. MR 1214142, DOI 10.1007/bf01083679
[Lie] S. Lie, Theorie der transformationsgruppen (Zweiter Abschnitt, unter Mitwirkung von Prof. Dr. Friederich Engel), Teubner, Leipzig, 1890.
André Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry 12 (1977), no. 2, 253–300 (French). MR 501133
[M] K. C. H. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, vol. 124, Cambridge University Press, 1987.
Kirill C. H. Mackenzie and Ping Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J. 73 (1994), no. 2, 415–452. MR 1262213, DOI 10.1215/S0012-7094-94-07318-3
Alan Weinstein, The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), no. 3, 523–557. MR 723816
Alan Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.) 16 (1987), no. 1, 101–104. MR 866024, DOI 10.1090/S0273-0979-1987-15473-5
Alan Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan 40 (1988), no. 4, 705–727. MR 959095, DOI 10.2969/jmsj/04040705
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
- [Bo]
- N. Bourbaki, Lie groups and Lie algebras, Part I, Springer-Verlag, Berlin and New York, 1989. MR 89k:17001
- [Br]
- J.-L. Brylinski, A differential complex for Poisson manifolds, J. Diff. Geom. 28 (1988), 93--114.MR 89m:58006
- [C]
- J. Conn, Normal forms for analytic Poisson structures, Ann. of Math. 119 (1984), 577--601; Normal forms for smooth Poisson structures, Ann. of Math. 121 (1985), 565--593.MR 86m:58050
- [D]
- P. M. Dirac, Lectures on quantum mechanics, Befer Graduate School Sci. Yeshiva Univ., New York, 1964.
- [FS]
- M. Flato and D. Sternheimer, Closedness of star products and cohomologies, Lie Theory and Geometry, in Honor of B. Kostant, Progress in Math. 123, Birkhäuser, New York, 1994. MR 1:327 536
- [H]
- R. Hermann, Cartan connections and the equivalence problem for geometric structures, Contributions to Differential Equations 3 (1964), 199--248. MR 29:2741
- [Kir]
- A. A. Kirillov, Local Lie algebras, Russian Math. Surveys 31 (1976), 57--76. MR 55:11304a
- [Kar]
- M. V. Karasev, Analogues of the objects of Lie group theory for nonlinear Poisson brackets, Math. USSR-Izv. 28 (1987), 497--527.
- [KM]
- M. V. Karasev and V. P. Maslov, Nonlinear Poisson brackets, geometry and quantization, Transl. Math. Monographs, vol. 119, Amer. Math. Soc., Providence, RI, 1993. MR 94a:58072
- [Lie]
- S. Lie, Theorie der transformationsgruppen (Zweiter Abschnitt, unter Mitwirkung von Prof. Dr. Friederich Engel), Teubner, Leipzig, 1890.
- [Lic]
- A. Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Diff. Geom. 12 (1977), 253--300.MR 58:18565
- [M]
- K. C. H. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, vol. 124, Cambridge University Press, 1987.
- [MX]
- K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J. 73 (1994), 415--452.MR 95B:58171
- [W1]
- A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom. 18 (1983), 523--557. MR 86i:58059
- [W2]
- ------, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.) 16 (1987), 101--104. MR 88c:58019
- [W3]
- ------, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan 40 (1988), 705--727. MR 90b:58091
- [W4]
- ------, Deformation quantization, Sém. Bourbaki, 46ème année, no. 789 (1993-1994), Asterisque 227 (1995), 389--409. MR 1:321 655
Review Information:
Reviewer:
Ping Xu
Affiliation:
The Pennsylvania State University
Email:
ping@math.psu.edu
Journal:
Bull. Amer. Math. Soc.
33 (1996), 255-261
DOI:
https://doi.org/10.1090/S0273-0979-96-00644-1
Review copyright:
© Copyright 1996
American Mathematical Society
