Poisson-Nijenhuis structures, truncated loop algebras and Sato's KP hierarchy.

The main point of this paper is to give a bi-Hamiltonian approach to the Kadomtsev-Petviashvili hierarchy of integrable nonlinear differential equations with two spatial variables (as well as Sato's generalizations of this hierarchy). Up to this time, bi-Hamiltonian structures have been displayed for various reductions of these hierarchies, namely the Gelʹfand-Dikiĭ hierarchies of integrable PDEs with one spatial variable. (Also, most finite-dimensional integrable systems have by now been cast in a bi-Hamiltonian setting.) By studying natural multi-Hamiltonian structures that arise on the direct sum of $n$ copies of the dual of a Lie algebra, and letting $n$ go to infinity appropriately, this paper demonstrates bi-Hamiltonian structures for the full Kadomtsev-Petviashvili-Sato hierarchies.