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Nonlinear Poisson brackets : geometry and quantization / M. V. Karasev, V. P. Maslov ; [transl. from the Russian by A. Sossinsky and M. Shishkova]

Auteur principal : Karasev, Mihail Vladimirovich, AuteurCo-auteur : Maslov, Viktor Pavlovich, 1930-, AuteurAuteur secondaire : Sosinskii, Aleksei Bronislavovich, 1937-, Traducteur • Shishkova, Maria, TraducteurType de document : MonographieCollection : Translations of mathematical monographs, 119Langue : anglais.Pays : Etats Unis.Éditeur : Providence : American Mathematical Society, 1993Description : 1 vol. (XI-366 p.) : ill. ; 26 cmISBN : 0821845969.ISSN : 0065-9282.Bibliographie : Bibliogr. p. 353-366.Sujet MSC : 58-02, Global analysis, analysis on manifolds, Research exposition (monographs, survey articles)
53D50, Differential geometry -- Symplectic geometry, contact geometry, Geometric quantization
81Q20, Quantum theory -- General mathematical topics and methods in quantum theory, Semiclassical techniques, including WKB and Maslov methods
58Jxx, Global analysis, analysis on manifolds, Partial differential equations on manifolds; differential operators
81S10, Quantum theory -- General quantum mechanics and problems of quantization, Geometry and quantization, symplectic methods
En-ligne : Zentralblatt | MathSciNet | AMS
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Bibliogr. p. 353-366

This book deals with two old mathematical problems. The first is the problem of constructing an analog of a Lie group for general nonlinear Poisson brackets. The second is the quantization problem for such brackets in the semiclassical approximation (which is the problem of exact quantization for the simplest classes of brackets). These problems are progressively coming to the fore in the modern theory of differential equations and quantum theory, since the approach based on constructions of algebras and Lie groups seems, in a certain sense, to be exhausted. The authors' main goal is to describe in detail the new objects that appear in the solution of these problems. Many ideas of algebra, modern differential geometry, algebraic topology, and operator theory are synthesized here. The authors prove all statements in detail, thus making the book accessible to graduate students. (source : AMS)

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