Author | Vaisman, Izu. author |
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Title | Lectures on the Geometry of Poisson Manifolds [electronic resource] / by Izu Vaisman |
Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1994 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-8495-2 |
Descript | VII, 206 p. online resource |
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 and biinvariant 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