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Product Code:  MMONO/119.S 
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Product Code:  MMONO/119.S.E 
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Softcover ISBN:  9780821887967 
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Product Code:  MMONO/119.S.B 
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Softcover ISBN:  9780821887967 
Product Code:  MMONO/119.S 
List Price:  $165.00 
MAA Member Price:  $148.50 
AMS Member Price:  $132.00 
eBook ISBN:  9781470416195 
Product Code:  MMONO/119.S.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 
Softcover ISBN:  9780821887967 
eBook ISBN:  9781470416195 
Product Code:  MMONO/119.S.B 
List Price:  $320.00 $242.50 
MAA Member Price:  $288.00 $218.25 
AMS Member Price:  $256.00 $194.00 

Book DetailsTranslations of Mathematical MonographsVolume: 119; 1993; 366 ppMSC: Primary 58; 81; 03; Secondary 16; 05
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.
ReadershipGraduate students and researchers.

Table of Contents

Chapters

Introduction

Chapter 1. Poisson manifolds

Chapter 2. Analog of the group operation for nonlinear Poisson brackets

Chapter 3. Poisson brackets in $R^{2n}$ and semiclassical approximation

Chapter 4. Asymptotic quantization


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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.
Graduate students and researchers.

Chapters

Introduction

Chapter 1. Poisson manifolds

Chapter 2. Analog of the group operation for nonlinear Poisson brackets

Chapter 3. Poisson brackets in $R^{2n}$ and semiclassical approximation

Chapter 4. Asymptotic quantization