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abstract: 'We put the Adler-Gelfand-Dickey approach to classical ${\mathcal{W}}$-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the KP hierarchy, together with its generalizations and reduction to the $N$-th KdV hierarchy, using the formal distribution calculus and the $\lambda$-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 (non-local) bi-Poisson structures of the matrix KP and the matrix $N$-th KdV hierarchies, and we prove integrability of the $N$-th matrix KdV hierarchy.'
address: 'Dip.to Matematica, Università La Sapienza, P.le Aldo Moro 2, 00185 Rome, Italy, desole@mat.uniroma1.it, Department of Mathematics, MIT, 77 Mass. Avenue, Cambridge, MA 02139, USA, kac@math.mit.edu, and SISSA, Via Bonomea 265, 34136 Trieste, Italy, dvaleri@sissa.it. '
author:
- 'Alberto De Sole, Victor G. Kac and Daniele Valeri'
title: 'Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras'
---

Introduction
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It is well known that classical ${\mathcal{W}}$-algebras play an important role in the theory of integrable bi-Hamiltonian equations. One of 