REDUCTION OF TODA LATTICE HIERARCHY TO GENERALIZED KDV HIERARCHIES AND THE 2-MATRIX MODEL

Toda lattice hierarchy and the associated matrix formulation of the 2M -boson KP hierarchies provide a framework for the Drinfeld-Sokolov red uction scheme realized through Hamiltonian action within the second KP Poisson bracket. By working with free currents, which Abelianize the second KP Hamiltonian structure, we are able to obtain a unified forma lism for the reduced SL(M + 1, M - k) KdV hierarchies interpolating be tween the ordinary KP and KdV hierarchies. The corresponding Lax opera tors are given as superdeterminants of graded SL(M + 1, M - k) matrice s in the diagonal gauge and we describe their bracket structure and fi eld content. In particular, we provide explicit free field representat ions of the associated W(M, M - k) Poisson bracket algebras generalisi ng the familiar nonlinear W-M+1 algebra. Discrete Backlund transformat ions for SL(M + 1, M - k) KdV are generated naturally from lattice tra nslations in the underlying Toda-like hierarchy. As an application we demonstrate the equivalence of the two-matrix string model to the SL(M + 1, 1) KdV hierarchy.