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

Type 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, Research exposition (monographs, survey articles) pertaining to global analysis53D50, Differential geometry - Symplectic geometry, contact geometry, Geometric quantization

81Q20, 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, General quantum mechanics and problems of quantization, Geometry and quantization, symplectic methodsEn-ligne : Zentralblatt | MathSciNet | AMS Item type: Monographie

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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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