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Bihamiltonian cohomology of KdV brackets. (English) Zbl 1362.37124
In this paper, using the spectral sequence method, the bi-Hamiltonian cohomology groups of the pencil of Poisson brackets of dispersionless KdV hierarchy is calculated. In particular, a conjecture of Luang Zhang about the vanishing of such cohomology groups is proved. In Section 2, the formalism of local functional polyvector fields and the definition of bi-Hamiltonian cohomology groups are given. In Section 3, some textbook material on spectral sequences that is used to study the problem is discussed. In Section 4, the filtration of the polynomial complex is introduced and the main theorems on its cohomology are proved by using the induced spectral sequence. The bi-Hamiltonian cohomology of the dispersionless KdV Poisson pencil is derived.

MSC:
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
20J06 Cohomology of groups
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References:
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