Lectures on the Geometry of Poisson Manifolds

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.

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