found a Poisson structure for these hierarchies, obtained as a Poisson algebra on a suitable space of scalar differential operators ($N$ being the order of the operators). A candidate for a second Poisson structure (whence a bi-Poisson structure) was conjectured by [@Adl79] and proved by [@GD78]. These Poisson structures are usually known as the *first and second Adler-Gelfand-Dickey (AGD) Poisson structures* and the second one is a special case of general classical ${\mathcal{W}}$-algebras.

Later on, all the $N$-th KdV hierarchies were “embedded” into one big hierarchy called the KP hierarchy, [@Sat81]. A Poisson structure for this hierarchy (which was a slight modification of the AGD Poisson structure) was suggested by [@Wat83], and [@Dic87] proved that there exists a second Poisson structure. In fact, it was shown by [@Rad87] that, for any $N\geq1$, there is a bi-Poisson structure for the KP hierarchy which induces the bi-Poisson structure for the $N$-th KdV hierarchy. We refer to the book of [@Dic03] for a detailed exposition of these topics and a large list of references.

The main goal of the present paper is to give a Poisson vertex algebra interpretation of the AGD approach to classical ${\mathcal{W}}$-algebras. This point of view greatly simplifies the 