Morita equivalence of Poisson manifolds. (English) Zbl 0746.58034

The author uses the analogy between Poisson manifolds and their symplectic realizations on the one side and associative algebras (in particular, \(C^*\)-algebras) and their representations on the other side. The notion of Morita equivalence of Poisson manifolds is introduced by analogy of Morita equivalence of algebras. It is proved that Morita equivalent Poisson manifolds have equivalent “categories” of complete symplectic realizations. The geometric invariants of Morita equivalence are also investigated; namely it is shown that for a Poisson manifold in which the characteristic foliations are trivial fibrations the variation lattice of symplectic structures along symplectic leaves is a complete Morita equivalence invariant.


37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
53D50 Geometric quantization
16D90 Module categories in associative algebras
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