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Lectures on the Geometry of Poisson Manifolds

Bibliographic Details
Main Author: Vaisman, Izu
Format: eBook
Language:English
Published: Basel Birkhäuser 1994, 1994
Edition:1st ed. 1994
Series:Progress in Mathematics
Subjects:
Geometry, Differential
Mathematical Physics
Manifolds And Cell Complexes
Manifolds (mathematics)
Differential Geometry
Theoretical, Mathematical And Computational Physics
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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
  • 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

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