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On deformation of Poisson manifolds of hydrodynamic type. (English) Zbl 1108.53044
The paper under review addresses a question of B. Dubrovin on formal deformations of certain infinite-dimensional Poisson manifolds. Specifically, one is interested in the manifold of all smooth mappings of the unit circle into \({\mathbb R}^n\), equipped with a Poisson bivector field of hydrodynamic type. The aforementioned question concerns the triviality of homogeneous formal deformations of these manifolds. The authors’ approach to this problem relies on the facts that the Poisson manifolds of hydrodynamic type are transversally constant and that the second Poisson-Lichnerowicz cohomology groups of these manifolds are essentially trivial in some sense.

MSC:
53D17 Poisson manifolds; Poisson groupoids and algebroids
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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