HAMILTONIAN STRUCTURES OF THE MULTI-BOSON KP HIERARCHIES, ABELIANIZATION AND LATTICE FORMULATION

We present a new form of the multi-boson reduction of KP hierarchy wit h Lax operator written in terms of boson fields abelianizing the secon d Hamiltonian structure. This extends the classical Miura transformati on and the Kupershmidt-Wilson theorem from the (m) KdV to the KP case. A remarkable relationship is uncovered between the higher Hamiltonian structures and the corresponding Miura transformations of KP hierarch y, on one hand, and the discrete integrable models living on refinemen ts of the original lattice connected with the underlying multi-matrix models, on the other hand. For the second KP Hamiltonian structure, wo rked out in details, this amounts to finding a series of representatio ns of the nonlinear W(infinity) algebra in terms of arbitrary finite n umber of canonical pairs of free fields.