Adler–Gelfand–Dickey Approach to Classical -Algebras Within the Theory of Poisson Vertex Algebras

We put the Adler-Gelfand-Dickey approach to classical W-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the Kadomtsev-Petviashvili (KP) hierarchy, together with its generalizations and reduction to the Nth Korteweg-de Vries (KdV) hierarchy, using the formal distribution calculus and the λ-bracket formalism. We apply the Lenard-Magri scheme to prove integrability of the corresponding hierarchies. We also give a simple proof of a theorem of Kupershmidt and Wilson in this framework. Based on this approach, we generalize all these results to the matrix case. In particular, we find (nonlocal) bi-Poisson structures of the matrix KP and the matrix Nth KdV hierarchies, and we prove integrability of the Nth matrix KdV hierarchy