The theory of Poisson Vertex Algebras (PVAs) [4] is a good framework to treat Hamiltonian partial differential equations. A PVA consists of a pair (A, {· λ ·}) of a differential algebra A and a bilinear operation called the λ-bracket. We extend the definition to the class of algebras A endowed with d ≥ 1 commuting derivations. We call this structure a multidimensional PVA: it is a suitable setting to study Hamiltonian PDEs with d spatial dimensions. We apply this theory to the study of symmetries and deformations of the Poisson brackets of hydrodynamic type for d = 2.

We compare three different ways of checking the Jacobi identity for weakly nonlocal Poisson brackets using the theory of distributions, pseudo-differential operators, and Poisson vertex algebras, respectively. We show that the three approaches lead to similar computations and same results. K E Y W O R D Smathematical physics, partial differential equations, solitons and integrable systems 412

Abstract. We compute the Poisson cohomology of a scalar Poisson bracket of Dubrovin-Novikov type with D independent variables. We find that the second and third cohomology groups are generically nonvanishing in D > 1. Hence, in contrast with the D = 1 case, the deformation theory in the multivariable case is non-trivial.

In this paper, we carry out the algebraic study of integrable differential-difference equations whose field variables take values in an associative (but not commutative) algebra. We adapt the Hamiltonian formalism to nonabelian difference Laurent polynomials and describe how to obtain a recursion operator from the Lax representation of an integrable nonabelian differential-difference system. As an application, we study a family of integrable equations: the nonabelian Narita–Itoh–Bogoyavlensky lattice, for which we construct their recursion operators and Hamiltonian operators and prove the locality of infinitely many commuting symmetries generated from their highly nonlocal recursion operators. Finally, we discuss the nonabelian version of several integrable difference systems, including the relativistic Toda chain and Ablowitz–Ladik lattice.

Based on the theory of Poisson vertex algebras we calculate skew-symmetry conditions and Jacobi identities for a class of third-order nonlocal operators of differential-geometric type. Hamiltonian operators within this class are defined by a Monge metric and a skew-symmetric two-form satisfying a number of differentialgeometric constraints. Complete classification results in the 2-component and 3component cases are obtained. MSC: 37K05, 37K10, 37K20, 37K25.