Classical W -algebras in higher dimensions are constructed. This is achieved by generalizing the classical Gel'fand-Dickey brackets to the commutative limit of the ring of classical pseudodifferential operators in arbitrary dimension. These W -algebras are the Poisson structures associated with a higher dimensional version of the Khokhlov-Zabolotskaya hierarchy (dispersionless KP-hierarchy). The two dimensional case is worked out explicitly and it is shown that the role of DiffS (1) is taken by the algebra of generators of local diffeomorphisms in two dimensions.

Classical W-algebras in higher dimensions have recently been constructed. In this letter we show that there is a finitely generated subalgebra which is isomorphic to the algebra of local diffeomorphisms in D dimensions. Moreover, there is a tower of infinitely many fields transforming under this subalgebra as symmetric tensorial one-densities. We also unravel a structure isomorphic to the Schouten symmetric bracket, providing a natural generalization of w∞ in higher dimensions.

It is well known that the centerless W 1+∞ algebra provides a hamiltonian structure for the KP hierarchy. In this letter we address the question whether the centerful version plays a similar rôle in any related integrable system. We find that, surprisingly enough, the centrally extended W 1+∞ algebra yields yet another Poisson structure for the same standard KP hierarchy. This is proven by explicit construction of the infinitely many new hamiltonians in closed form. †