2005/06, 4th semester

Course Materials
From Poisson to quantum geometry

One way of approaching the study of noncommutative geometry is to consider it as a deformation of the usual commutative geometry. In this approach, the infinitesimal part of the deformation, i.e., the first order in the deformation parameter h, carries substantial information on global deformations. This infinitesimal part turns out to be (equivalent to) a Poisson bracket on the commutative algebra, and thus endows the underlying manifold with the structure of a Poisson manifold. In many different situations, it turns out that the deformation is completely determined by its infinitesimal part. Therefore, it is reasonable to seek understanding of any quantum behaviour of a noncommutative algebra (insofar as it differs from the classical behaviour) in terms of a Poisson geometry on the underlying manifold. A basic knowledge of differential geometry, including de Rham theory, will be helpful. Prerequisites on Hopf algebras and on Hochschild and cyclic homologies may be provided if participants request it.

This lecture course is intended to be an introduction to this geometrical approach to noncommutative algebras, mainly focused on quantization of Lie groups and their homogeneous spaces. We will start by introducing the basic theory of Poisson manifolds, centering afterwards on the different notions of equivalence in the Poisson category (Poisson diffeomorphism, Poisson Morita equivalence, gauge equivalence), and the corresponding invariants (Poisson homology and cohomology, the modular class, contravariant connections and Poisson K-theory). A few examples of invariant computations, a process known to be usually quite hard, will be given. We will then move on to multiplicative structures on Lie groups, focusing on the theory of compact Poisson-Lie groups. Special emphasis will be put on the construction and classification of Poisson homogeneous spaces, together with explicit examples. The next step will be to set up a parallel construction in the noncommutative setting (quantum groups). This will show the role played by the analogy with the Poisson category in finding the right approach to quantum homogeneous spaces (constructing them both from global and infinitesimal data). The final aim will be to show the connections between noncommutative homological invariants and Poisson invariants, reviewing the Feng-Tsygan computation of Hochschild and cyclic homology of quantum groups in terms of Poisson homology, discussing the work of Eli Hawkins on Poisson obstructions to noncommutative spectral triples and, if time permits, entering the largely unexplored field of Poisson invariants underlying quantum vector bundles.

From Poisson to quantum geometry
Nicola Ciccoli
Notes by P. Witkowski
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Table of contents
  1. Poisson geometry
    1. Poisson algebra
    2. Poisson manifolds
    3. The sharp map
    4. The symplectic foliation
  2. Schouten-Nijenhuis bracket
    1. Lie-Poisson bracket
    2. Schouten-Nijenhuis bracket
    3. Poisson homology
  3. Poisson maps
    1. Poisson maps
    2. Poisson submanifolds
    3. Coinduced Poisson structures
    4. Completeness
  4. Poisson cohomology
    1. Modular class
    2. Computations for Poisson cohomology
  5. Poisson homology
    1. Poisson homology and modulatr class
  6. Coisotropic submanifolds
    1. Poisson Morita equivalence
    2. Dirac structures
  7. Poisson Lie groups
    1. Poisson Lie groups
    2. Lie bialgebras
    3. Manin triples
  8. Poisson actions
    1. Poisson actions
    2. Poisson homogeneous spaces
    3. Dressing actions
  9. Quantization
    1. Introduction
    2. Duality
    3. Local, global, special quantization
    4. Real structures
    5. Dictionary
    6. Quantum subgroups
    7. Quantum homogeneous spaces
    8. Coisotropic creed
Designed by: Pawel Witkowski