Lectures on the Geometry of Poisson Manifolds

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Birkhäuser, Dec 6, 2012 - Mathematics - 206 pages
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This book is addressed to graduate students and researchers in the fields of mathematics and physics who are interested in mathematical and theoretical physics, differential geometry, mechanics, quantization theories and quantum physics, quantum groups etc., and who are familiar with differentiable and symplectic manifolds. The aim of the book is to provide the reader with a monograph that enables him to study systematically basic and advanced material on the recently developed theory of Poisson manifolds, and that also offers ready access to bibliographical references for the continuation of his study. Until now, most of this material was dispersed in research papers published in many journals and languages. The main subjects treated are the Schouten-Nijenhuis bracket; the generalized Frobenius theorem; the basics of Poisson manifolds; Poisson calculus and cohomology; quantization; Poisson morphisms and reduction; realizations of Poisson manifolds by symplectic manifolds and by symplectic groupoids and Poisson-Lie groups. The book unifies terminology and notation. It also reports on some original developments stemming from the author's work, including new results on Poisson cohomology and geometric quantization, cofoliations and biinvariant Poisson structures on Lie groups.
 

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Contents

Acknowledgement
1
The symplectic foliation of a Poisson manifold
19
Examples of Poisson manifolds
31
Poisson calculus
41
Poisson cohomology
63
An introduction to quantization
83
Poisson morphisms coinduced structures reduction
97
Symplectic realizations of Poisson manifolds
115
Realizations of Poisson manifolds
134
PoissonLie groups
161
References
189
Index 203
202
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