Lectures on the geometry of Poisson manifolds

摘要: 0 Introduction.- 1 The Poisson bivector and the Schouten-Nijenhuis bracket.- 1.1 bivector.- 1.2 1.3 Coordinate expressions.- 1.4 Koszul formula applications.- 1.5 Miscellanea.- 2 symplectic foliation of a manifold.- 2.1 General distributions foliations.- 2.2 Involutivity integrability.- 2.3 case manifolds.- 3 Examples 3.1 Structures on ?n. Lie-Poisson structures.- 3.2 Dirac brackets.- 3.3 Further examples.- 4 calculus.- 4.1 bracket 1-forms.- 4.2 contravariant exterior differentiations.- 4.3 regular case.- 4.4 Cofoliations.- 4.5 Contravariant derivatives vector bundles.- 4.6 More 5 cohomology.- 5.1 Definition general properties.- 5.2 Straightforward inductive computations.- 5.3 spectral sequence 5.4 homology.- 6 An introduction to quantization.- 6.1 Prequantization.- 6.2 Quantization.- 6.3 Prequantization representations.- 6.4 Deformation 7 morphisms, coinduced structures, reduction.- 7.1 Properties mappings.- 7.2 Reduction 7.3 Group actions momenta.- 7.4 8 Symplectic realizations 8.1 Local realizations.- 8.2 Dual pairs 8.3 Isotropic 8.4 nets.- 9 Realizations manifolds by groupoids.- 9.1 9.2 Lie groupoid structures T*G.- 9.3 9.4 algebroids integrability 9.5 results.- 10 Poisson-Lie groups.- 10.1 biinvariant 10.2 Characteristic properties 10.3 algebra group.- 10.4 Yang-Baxter equations.- 10.5 Manin triples.- 10.6 Actions dressing transformations.- References.