01808nas a2200121   4500008004100000245011400041210006900155260001000224520121600234653008901450100001901539856012801558       2015                        en  d00aMultidimensional Poisson Vertex Algebras and Poisson cohomology of Hamiltonian operators of hydrodynamic type0 aMultidimensional Poisson Vertex Algebras and Poisson cohomology   bSISSA3 aThe Poisson brackets of hydrodynamic type, also called Dubrovin-Novikov brackets, constitute the Hamiltonian structure of a broad class of evolutionary PDEs, that are ubiquitous in the theory of Integrable Systems, ranging from Hopf equation to the principal hierarchy of a Frobenius manifold. They can be regarded as an analogue of the classical
Poisson brackets, defined on an infinite dimensional space of maps Σ → M between two manifolds. Our main problem is the study of Poisson-Lichnerowicz cohomology of such space when dim Σ > 1. We introduce the notion of multidimensional Poisson Vertex Algebras, generalizing and adapting the theory by A. Barakat, A. De Sole, and V. Kac [Poisson Vertex Algebras in the theory of Hamiltonian equations, 2009]; within this framework we explicitly compute the first nontrivial cohomology groups for an arbitrary Poisson bracket of hydrodynamic type, in the case dim Σ = dim M = 2. For the case of the so-called scalar brackets, namely the ones for which dim M = 1, we give a complete description on their Poisson–Lichnerowicz cohomology. From this computations it
follows, already in the particular case dim Σ = 2, that the cohomology is infinite dimensional.10aPoisson Vertex Algebras, Poisson brackets, Hamiltonian operators, Integrable Systems1 aCasati, Matteo  uhttps://www.math.sissa.it/publication/multidimensional-poisson-vertex-algebras-and-poisson-cohomology-hamiltonian-operators