The Classical Limit of W-Algebras

We define and compute explicitly the classical limit of the realizations of $W_n$ appearing as hamiltonian structures of generalized KdV hierarchies. The classical limit is obtained by taking the commutative limit of the ring of pseudodifferential operators. These algebras---denoted $w_n$---have free field realizations in which the generators are given by the elementary symmetric polynomials in the free fields. We compute the algebras explicitly and we show that they are all reductions of a new algebra $w_{\rm KP}$, which is proposed as the universal classical $W$-algebra for the $w_n$ series. As a deformation of this algebra we also obtain $w_{1+\infty}$, the classical limit of $W_{1+\infty}$.