Lectures on the Geometry of Poisson Manifolds (Progress in Mathematics)
Vaisman, Izu
ISBN 10: 3034896492 ISBN 13: 9783034896498
Editore: Birkhäuser, 2012
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Everybody having even the slightest interest in analytical mechanics remembers having met there the Poisson bracket of two functions of 2n variables (pi, qi) f g ~(8f8g 8 8 ) (0.1) {f,g} = L... ~[ji - [ji~ , ;=1 p, q q p, and the fundamental role it plays in that field. In modern works, this bracket is derived from a symplectic structure, and it appears as one of the main in gredients of symplectic manifolds. In fact, it can even be taken as the defining clement of the structure (e.g., [TIl]). But, the study of some mechanical sys tems, particularly systems with symmetry groups or constraints, may lead to more general Poisson brackets. Therefore, it was natural to define a mathematical structure where the notion of a Poisson bracket would be the primary notion of the theory, and, from this viewpoint, such a theory has been developed since the early 19708, by A. Lichnerowicz, A. Weinstein, and many other authors (see the references at the end of the book). But, it has been remarked by Weinstein [We3] that, in fact, the theory can be traced back to S. Lie himself [Lie].
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"The book serves well as an introduction and an overview of the subject and a long list of references helps with further study."
-- Zbl. Math.
"The book is well done...should be an essential purchase for mathematics libraries and is likely to be a standard reference for years to come, providing an introduction to an attractive area of further research." -- Mathematical Reviews
"The importance and actuality of the subject, as well as the very rigorous and didactic presentation of the content, make out of this book a valuable contribution to current mathematics. The book is intended first of all to mathematicians, but it can be interesting also for a wide circle of readers, including mechanicists and physicists." -- Mathematica
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Titolo: Lectures on the Geometry of Poisson ...
Casa editrice: Birkhäuser
Data di pubblicazione: 2012
Legatura: Brossura
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Lectures on the Geometry of Poisson Manifolds
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Condizione: New. Dieser Artikel ist ein Print on Demand Artikel und wird nach Ihrer Bestellung fuer Sie gedruckt. 0 Introduction.- 1 The Poisson bivector and the Schouten-Nijenhuis bracket.- 1.1 The Poisson bivector.- 1.2 The Schouten-Nijenhuis bracket.- 1.3 Coordinate expressions.- 1.4 The Koszul formula and applications.- 1.5 Miscellanea.- 2 The symplectic foliation . Codice articolo 4319462
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Taschenbuch. Condizione: Neu. Lectures on the Geometry of Poisson Manifolds | Izu Vaisman | Taschenbuch | vii | Englisch | 2012 | Birkhäuser | EAN 9783034896498 | Verantwortliche Person für die EU: Springer Basel AG in Springer Science + Business Media, Heidelberger Platz 3, 14197 Berlin, juergen[dot]hartmann[at]springer[dot]com | Anbieter: preigu. Codice articolo 105706216
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Taschenbuch. Condizione: Neu. This item is printed on demand - Print on Demand Titel. Neuware -0 Introduction.- 1 The Poisson bivector and the Schouten-Nijenhuis bracket.- 1.1 The Poisson bivector.- 1.2 The Schouten-Nijenhuis bracket.- 1.3 Coordinate expressions.- 1.4 The Koszul formula and applications.- 1.5 Miscellanea.- 2 The symplectic foliation of a Poisson manifold.- 2.1 General distributions and foliations.- 2.2 Involutivity and integrability.- 2.3 The case of Poisson manifolds.- 3 Examples of Poisson manifolds.- 3.1 Structures on n. Lie-Poisson structures.- 3.2 Dirac brackets.- 3.3 Further examples.- 4 Poisson calculus.- 4.1 The bracket of 1-forms.- 4.2 The contravariant exterior differentiations.- 4.3 The regular case.- 4.4 Cofoliations.- 4.5 Contravariant derivatives on vector bundles.- 4.6 More brackets.- 5 Poisson cohomology.- 5.1 Definition and general properties.- 5.2 Straightforward and inductive computations.- 5.3 The spectral sequence of Poisson cohomology.- 5.4 Poisson homology.- 6 An introduction to quantization.- 6.1 Prequantization.- 6.2 Quantization.- 6.3 Prequantization representations.- 6.4 Deformation quantization.- 7 Poisson morphisms, coinduced structures, reduction.- 7.1 Properties of Poisson mappings.- 7.2 Reduction of Poisson structures.- 7.3 Group actions and momenta.- 7.4 Group actions and reduction.- 8 Symplectic realizations of Poisson manifolds.- 8.1 Local symplectic realizations.- 8.2 Dual pairs of Poisson manifolds.- 8.3 Isotropic realizations.- 8.4 Isotropic realizations and nets.- 9 Realizations of Poisson manifolds by symplectic groupoids.- 9.1 Realizations of Lie-Poisson structures.- 9.2 The Lie groupoid and symplectic structures of T\*G.- 9.3 General symplectic groupoids.- 9.4 Lie algebroids and the integrability of Poisson manifolds.- 9.5 Further integrability results.- 10 Poisson-Lie groups.- 10.1 Poisson-Lie andbiinvariant structures on Lie groups.- 10.2 Characteristic properties of Poisson-Lie groups.- 10.3 The Lie algebra of a Poisson-Lie group.- 10.4 The Yang-Baxter equations.- 10.5 Manin triples.- 10.6 Actions and dressing transformations.- References.Springer Nature c/o IBS, Benzstrasse 21, 48619 Heek 220 pp. Englisch. Codice articolo 9783034896498
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ISBN 13: 9783034896498
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Taschenbuch. Condizione: Neu. This item is printed on demand - it takes 3-4 days longer - Neuware -Everybody having even the slightest interest in analytical mechanics remembers having met there the Poisson bracket of two functions of 2n variables (pi, qi) f g ~(8f8g 8 8 ) (0.1) {f,g} = L. ~[ji - [ji~ , ;=1 p, q q p, and the fundamental role it plays in that field. In modern works, this bracket is derived from a symplectic structure, and it appears as one of the main in gredients of symplectic manifolds. In fact, it can even be taken as the defining clement of the structure (e.g., [TIl]). But, the study of some mechanical sys tems, particularly systems with symmetry groups or constraints, may lead to more general Poisson brackets. Therefore, it was natural to define a mathematical structure where the notion of a Poisson bracket would be the primary notion of the theory, and, from this viewpoint, such a theory has been developed since the early 19708, by A. Lichnerowicz, A. Weinstein, and many other authors (see the references at the end of the book). But, it has been remarked by Weinstein [We3] that, in fact, the theory can be traced back to S. Lie himself [Lie]. 206 pp. Englisch. Codice articolo 9783034896498
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Taschenbuch. Condizione: Neu. Druck auf Anfrage Neuware - Printed after ordering - Everybody having even the slightest interest in analytical mechanics remembers having met there the Poisson bracket of two functions of 2n variables (pi, qi) f g ~(8f8g 8 8 ) (0.1) {f,g} = L. ~[ji - [ji~ , ;=1 p, q q p, and the fundamental role it plays in that field. In modern works, this bracket is derived from a symplectic structure, and it appears as one of the main in gredients of symplectic manifolds. In fact, it can even be taken as the defining clement of the structure (e.g., [TIl]). But, the study of some mechanical sys tems, particularly systems with symmetry groups or constraints, may lead to more general Poisson brackets. Therefore, it was natural to define a mathematical structure where the notion of a Poisson bracket would be the primary notion of the theory, and, from this viewpoint, such a theory has been developed since the early 19708, by A. Lichnerowicz, A. Weinstein, and many other authors (see the references at the end of the book). But, it has been remarked by Weinstein [We3] that, in fact, the theory can be traced back to S. Lie himself [Lie]. Codice articolo 9783034896498
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