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An infinite number of free field realizations of the universal nonlinear$\hat{W}_{\infty}^{(N)}$ ($\hat{W}_{1+\infty}^{(N)}$) algebras, which areidentical to the KP Hamiltonian structures, are obtained in terms of $p$ plus$q$ scalars of different signatures with $p-q=N$. They are generalizations ofthe Miura transformation, and naturally give rise to the modified KPhierarchies via corresponding realizations of the latter. Their characteristicLie-algebraic origin is shown to be the graded $SL(p,q)$.Comment: 10p, (plain LaTex), Modified KP Hierarchies and Graded Lie Algebra

Many boson realizations of universal nonlinearW ?-algebras, modified KP hierarchies, and graded lie algebras