MULTICOMPONENT KDV HIERARCHY, V-ALGEBRA AND NON-ABELIAN TODA THEORY

We prove the recently conjectured relation between the 2 x 2-matrix di fferential operator L = partial derivative 2 - U and a certain nonline ar and nonlocal Poisson bracket algebra (V-algebra), containing a Vira soro subalgebra, which appeared in the study of a non-Abelian Toda fie ld theory. In particular, we show that this V-algebra is precisely giv en by the second Gelfand-Dikii bracket associated with L. The Miura tr ansformation that relates the second to the first Gelfand-Dikii bracke t is given. The two Gelfand-Dikii brackets are also obtained from the associated (integro-) differential equation satisfied by fermion bilin ears. The asymptotic expansion of the resolvent of (L - xi)PSI = 0 is studied and its coefficients R(l) yield an infinite sequence of Hamilt onians with mutually vanishing Poisson brackets. We recall how this le ads to a matrix KdV hierarchy, which here are flow equations for the t hree component fields T, V+, V- of U. For V+/- = 0, they reduce to the ordinary KdV hierarchy. The corresponding matrix mKdV equations are a lso given, as well as the relation to the pseudo-differential operator approach. Most of the results continue to hold if U is a Hermitian n x n matrix. Conjectures are made about n x n-matrix, mth-order differe ntial operators L and associated V(n,m)-algebras.