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Hardcover ISBN:  9780821844953 
Product Code:  SURV/151 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470413781 
Product Code:  SURV/151.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9780821844953 
eBook ISBN:  9781470413781 
Product Code:  SURV/151.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 151; 2008; 299 ppMSC: Primary 22; 43; 46; 53; 57; 78; 80
The threedimensional Heisenberg group, being a quite simple noncommutative Lie group, appears prominently in various applications of mathematics. The goal of this book is to present basic geometric and algebraic properties of the Heisenberg group and its relation to other important mathematical structures (the skew field of quaternions, symplectic structures, and representations) and to describe some of its applications. In particular, the authors address such subjects as signal analysis and processing, geometric optics, and quantization. In each case, the authors present necessary details of the applied topic being considered.
With no prerequisites beyond the standard mathematical curriculum, this book manages to encompass a large variety of topics being easily accessible in its fundamentals. It can be useful to students and researchers working in mathematics and in applied mathematics.
ReadershipGraduate students and research mathematicians interested in the use of analysis on Heisenberg groups to various problems in pure and applied mathematics.

Table of Contents

Chapters

1. The skew field of quaternions

2. Elements of the geometry of $S^3$, Hopf bundles and spin representations

3. Internal variables of singularity free vector fields in a Euclidean space

4. Isomorphism classes, Chern classes and homotopy classes of singularity free vector fields in 3space

5. Heisenberg algebras, Heisenberg groups, Minkowski metrics, Jordan algebras and SL$(2,\mathbb {C})$

6. The Heisenberg group and natural $C*$algebras of a vector field in 3space

7. The Schrödinger representation and the metaplectic representation

8. The Heisenberg group: A basic geometric background of signal analysis and geometric optics

9. Quantization of quadratic polynomials

10. Field theoretic Weyl quantization of a vector field in 3space

11. Thermodynamics, geometry and the Heisenberg group by Serge Preston


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The threedimensional Heisenberg group, being a quite simple noncommutative Lie group, appears prominently in various applications of mathematics. The goal of this book is to present basic geometric and algebraic properties of the Heisenberg group and its relation to other important mathematical structures (the skew field of quaternions, symplectic structures, and representations) and to describe some of its applications. In particular, the authors address such subjects as signal analysis and processing, geometric optics, and quantization. In each case, the authors present necessary details of the applied topic being considered.
With no prerequisites beyond the standard mathematical curriculum, this book manages to encompass a large variety of topics being easily accessible in its fundamentals. It can be useful to students and researchers working in mathematics and in applied mathematics.
Graduate students and research mathematicians interested in the use of analysis on Heisenberg groups to various problems in pure and applied mathematics.

Chapters

1. The skew field of quaternions

2. Elements of the geometry of $S^3$, Hopf bundles and spin representations

3. Internal variables of singularity free vector fields in a Euclidean space

4. Isomorphism classes, Chern classes and homotopy classes of singularity free vector fields in 3space

5. Heisenberg algebras, Heisenberg groups, Minkowski metrics, Jordan algebras and SL$(2,\mathbb {C})$

6. The Heisenberg group and natural $C*$algebras of a vector field in 3space

7. The Schrödinger representation and the metaplectic representation

8. The Heisenberg group: A basic geometric background of signal analysis and geometric optics

9. Quantization of quadratic polynomials

10. Field theoretic Weyl quantization of a vector field in 3space

11. Thermodynamics, geometry and the Heisenberg group by Serge Preston