Lectures on the Geometry of Poisson Manifolds

This book is addressed to graduate students and researchers in the fields of mathematics and physics who are interested in mathematical and theoretical physics, differential geometry, mechanics, quantization theories and quantum physics, ...

Author: Izu Vaisman

Publisher: Birkhäuser

ISBN: 9783034884952

Category: Mathematics

Page: 206

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This book is addressed to graduate students and researchers in the fields of mathematics and physics who are interested in mathematical and theoretical physics, differential geometry, mechanics, quantization theories and quantum physics, quantum groups etc., and who are familiar with differentiable and symplectic manifolds. The aim of the book is to provide the reader with a monograph that enables him to study systematically basic and advanced material on the recently developed theory of Poisson manifolds, and that also offers ready access to bibliographical references for the continuation of his study. Until now, most of this material was dispersed in research papers published in many journals and languages. The main subjects treated are the Schouten-Nijenhuis bracket; the generalized Frobenius theorem; the basics of Poisson manifolds; Poisson calculus and cohomology; quantization; Poisson morphisms and reduction; realizations of Poisson manifolds by symplectic manifolds and by symplectic groupoids and Poisson-Lie groups. The book unifies terminology and notation. It also reports on some original developments stemming from the author's work, including new results on Poisson cohomology and geometric quantization, cofoliations and biinvariant Poisson structures on Lie groups.

Lectures on Poisson Geometry

MR1096599 [140] Izu Vaisman, On the geometric quantization of Poisson manifolds, J. Math. Phys. 32 (1991), no. 12, 3339–3345, DOI 10.1063/1.529446. MR1137387 [141] Izu Vaisman, Lectures on the geometry of Poisson manifolds, ...

Author: Marius Crainic

Publisher: American Mathematical Soc.

ISBN: 9781470466671

Category: Education

Page: 479

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This excellent book will be very useful for students and researchers wishing to learn the basics of Poisson geometry, as well as for those who know something about the subject but wish to update and deepen their knowledge. The authors' philosophy that Poisson geometry is an amalgam of foliation theory, symplectic geometry, and Lie theory enables them to organize the book in a very coherent way. —Alan Weinstein, University of California at Berkeley This well-written book is an excellent starting point for students and researchers who want to learn about the basics of Poisson geometry. The topics covered are fundamental to the theory and avoid any drift into specialized questions; they are illustrated through a large collection of instructive and interesting exercises. The book is ideal as a graduate textbook on the subject, but also for self-study. —Eckhard Meinrenken, University of Toronto

Poisson Geometry in Mathematics and Physics

[16] Lu, J. H. and Yakimov, M., Group orbits and regular partitions of Poisson manifolds, math.SG/0609732. [17] Mackenzie, K., Lie groupoids and Lie algebroids in differential geometry, London Mathematical Society Lecture Note Series, ...

Author: Giuseppe Dito

Publisher: American Mathematical Soc.

ISBN: 9780821844236

Category: Mathematics

Page: 311

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This volume is a collection of articles by speakers at the conference ""Poisson 2006: Poisson Geometry in Mathematics and Physics"", which was held June 5-9, 2006, in Tokyo, Japan. Poisson 2006 was the fifth in a series of international conferences on Poisson geometry that are held once every two years. The aim of these conferences is to bring together mathematicians and mathematical physicists who work in diverse areas but have common interests in Poisson geometry. The program for Poisson 2006 was remarkable for the overlap of topics that included deformation quantization, generalized complex structures, differentiable stacks, normal forms, and group-valued moment maps and reduction. The articles represent current research in Poisson geometry and should be valuable to anyone interested in Poisson geometry, symplectic geometry, and mathematical physics. This volume also contains lectures by the principal speakers of the three-day school held at Keio University that preceded Poisson 2006.

The Breadth of Symplectic and Poisson Geometry

W. M. Tulczyjew, Geometric Formulation of Physical Theories, Bibliopolis, Napoli, 1989. I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Birkhäuser, Basel, Boston, Berlin, 1994. A. Weinstein, The local structure of Poisson ...

Author: Jerrold E. Marsden

Publisher: Springer Science & Business Media

ISBN: 9780817644192

Category: Mathematics

Page: 654

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* The invited papers in this volume are written in honor of Alan Weinstein, one of the world’s foremost geometers * Contributions cover a broad range of topics in symplectic and differential geometry, Lie theory, mechanics, and related fields * Intended for graduate students and working mathematicians, this text is a distillation of prominent research and an indication of future trends in geometry, mechanics, and mathematical physics

Quantum Algebras and Poisson Geometry in Mathematical Physics

( Vai ] I. Vaisman , Lectures on the Geometry of Poisson Manifolds , Progress in Math . , Vol . 118 , Birkhäuser , Boston , 1994 . [ Vaz ) Coupling Poisson and Jacobi structures on foliated manifolds , J. of Geometric Methods in Modern ...

Author: Mikhail Vladimirovich Karasev

Publisher: American Mathematical Soc.

ISBN: 0821840401

Category: Mathematics

Page: 277

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This collection presents new and interesting applications of Poisson geometry to some fundamental well-known problems in mathematical physics. The methods used by the authors include, in addition to advanced Poisson geometry, unexpected algebras with non-Lie commutation relations, nontrivial (quantum) Kahlerian structures of hypergeometric type, dynamical systems theory, semiclassical asymptotics, etc.

Poisson Structures

Geometric formulations of physical theories, volume 11 of Monographs and Textbooks in Physical Science. Lecture Notes. Bibliopolis, Naples, 1989. ... Lectures on the geometry of Poisson manifolds, volume 118 of Progress in Mathematics.

Author: Camille Laurent-Gengoux

Publisher: Springer Science & Business Media

ISBN: 9783642310904

Category: Mathematics

Page: 464

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Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson Structures is the first book that offers a comprehensive introduction to the theory, as well as an overview of the different aspects of Poisson structures. The first part covers solid foundations, the central part consists of a detailed exposition of the different known types of Poisson structures and of the (usually mathematical) contexts in which they appear, and the final part is devoted to the two main applications of Poisson structures (integrable systems and deformation quantization). The clear structure of the book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researchers who are interested in an introduction to the many facets and applications of Poisson structures.​

Introduction to Symplectic Geometry

American Mathematical Society, Providence (2014) [F3 Vai] Vaisman, I.: Lectures on the Geometry of Poisson Manifolds. Progress in Mathematics, vol. 118. Birkhäuser, Basel, Boston, Berlin (1994) [F3 W1] Weinstein, A.: The local structure ...

Author: Jean-Louis Koszul

Publisher: Springer

ISBN: 9789811339875

Category: Science

Page: 121

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This introductory book offers a unique and unified overview of symplectic geometry, highlighting the differential properties of symplectic manifolds. It consists of six chapters: Some Algebra Basics, Symplectic Manifolds, Cotangent Bundles, Symplectic G-spaces, Poisson Manifolds, and A Graded Case, concluding with a discussion of the differential properties of graded symplectic manifolds of dimensions (0,n). It is a useful reference resource for students and researchers interested in geometry, group theory, analysis and differential equations.This book is also inspiring in the emerging field of Geometric Science of Information, in particular the chapter on Symplectic G-spaces, where Jean-Louis Koszul develops Jean-Marie Souriau's tools related to the non-equivariant case of co-adjoint action on Souriau’s moment map through Souriau’s Cocycle, opening the door to Lie Group Machine Learning with Souriau-Fisher metric.

Geometric Models for Noncommutative Algebras

[ 156 ] Strătilă , Ş . , and Zsidó , L. , Lectures on von Neumann Algebras , revision of the 1975 original , translated from the ... [ 162 ] Vaisman , I. , Lectures on the Geometry of Poisson Manifolds , Birkhäuser , Basel , 1994 .

Author: Ana Cannas da Silva

Publisher: American Mathematical Soc.

ISBN: 0821809520

Category: Mathematics

Page: 184

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The volume is based on a course, ``Geometric Models for Noncommutative Algebras'' taught by Professor Weinstein at Berkeley. Noncommutative geometry is the study of noncommutative algebras as if they were algebras of functions on spaces, for example, the commutative algebras associated to affine algebraic varieties, differentiable manifolds, topological spaces, and measure spaces. In this work, the authors discuss several types of geometric objects (in the usual sense of sets with structure) that are closely related to noncommutative algebras. Central to the discussion are symplectic and Poisson manifolds, which arise when noncommutative algebras are obtained by deforming commutative algebras. The authors also give a detailed study of groupoids (whose role in noncommutative geometry has been stressed by Connes) as well as of Lie algebroids, the infinitesimal approximations to differentiable groupoids. Featured are many interesting examples, applications, and exercises. The book starts with basic definitions and builds to (still) open questions. It is suitable for use as a graduate text. An extensive bibliography and index are included.

Geometric Methods in Physics

Journal of Differential Geometry, 66 (2004). [5] Drinfeld, V.G., Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical ... [20] Vaisman, I., Lectures on the Geometry of Poisson Manifolds.

Author: Piotr Kielanowski

Publisher: Springer

ISBN: 9783319062488

Category: Mathematics

Page: 290

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The Białowieża Workshops on Geometric Methods in Physics, which are hosted in the unique setting of the Białowieża natural forest in Poland, are among the most important meetings in the field. Every year some 80 to 100 participants from both the mathematics and physics world join to discuss new developments and to exchange ideas. The current volume was produced on the occasion of the 32nd meeting in 2013. It is now becoming a tradition that the Workshop is followed by a School on Geometry and Physics, which consists of advanced lectures for graduate students and young researchers. Selected speakers at the 2013 Workshop were asked to contribute to this book, and their work was supplemented by additional review articles. The selection shows that, despite its now long tradition, the workshop remains at the cutting edge of research. The 2013 Workshop also celebrated the 75th birthday of Daniel Sternheimer, and on this occasion the discussion mainly focused on his contributions to mathematical physics such as deformation quantization, Poisson geometry, symplectic geometry and non-commutative differential geometry.