**Translations of Mathematical Monographs**

1993;
366 pp;
Softcover

MSC: Primary 58; 81; 03;
Secondary 16; 05

**Print ISBN: 978-0-8218-8796-7
Product Code: MMONO/119.S**

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

**Electronic ISBN: 978-1-4704-1619-5
Product Code: MMONO/119.S.E**

List Price: $115.00

AMS Member Price: $92.00

MAA Member Price: $103.50

# Nonlinear Poisson Brackets: Geometry and Quantization

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*M. V. Karasev; V. P. Maslov*

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

# Table of Contents

## Nonlinear Poisson Brackets: Geometry and Quantization

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Introduction 114
- Chapter 1. Poisson manifolds 720
- Chapter 2. Analog of the group operation for nonlinear Poisson brackets 7588
- Chapter 3. Poisson brackets in 𝑅²ⁿ and semi-classical approximation 139152
- Chapter 4. Asymptotic quantization 185198
- Appendix 1. Formulas of noncommutative analysis 265278
- Appendix 2. Calculus of symbols and commutation relations 297310
- References 353366
- Back Cover Back Cover1382