Construction of KP Hierarchies in Terms of Finite Number of Fields and their Abelianization

The 2 M -boson representations of KP hierarchy are constructed in terms of M mutually independent two-boson KP representations for arbitrary number M . Our construction establishes the multi-boson representations of KP hierarchy as consistent Poisson reductions of standard KP hierarchy within the R -matrix scheme. As a byproduct we obtain a complete description of any finitely-many-field formulation of KP hierarchy in terms of Darboux coordinates with respect to the first Hamiltonian structure. This results in a series of representations of W 1 + ∞ algebra made out of arbitrary even number of boson fields.