Translations of Mathematical Monographs 1993; 366 pp; softcover Volume: 119 ISBN10: 0821887963 ISBN13: 9780821887967 List Price: US$115 Member Price: US$92 Order Code: MMONO/119.S
 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. Readership Graduate students and researchers. Table of Contents  Poisson manifolds
 Analog of the group operation for nonlinear Poisson brackets
 Poisson brackets in \(\mathbb R^{2n}\) and semiclassical approximation
 Asymptotic quantization
 Appendix I: Formulas of noncommutative analysis
 Appendix II: Calculus of symbols and commutation relations
 References
