The (N,M)th Korteweg–de Vries hierarchy and the associated W-algebra

We discuss a differential integrable hierarchy, which we call the (N, M)$--thKdV hierarchy, whose Lax operator is obtained by properly adding $M$pseudo--differential terms to the Lax operator of the N--th KdV hierarchy. Thisnew hierarchy contains both the higher KdV hierarchy and multi--fieldrepresentation of KP hierarchy as sub--systems and naturally appears inmulti--matrix models. The N+2M-1 coordinates or fields of this hierarchysatisfy two algebras of compatible Poisson brackets which are {\it local} and{\it polynomial}. Each Poisson structure generate an extended W_{1+\infty} andW_\infty algebra, respectively. We call W(N, M) the generating algebra of theextended W_\infty algebra. This algebra, which corresponds with the secondPoisson structure, shares many features of the usual $W_N$ algebra. We showthat there exist M distinct reductions of the (N, M)--th KdV hierarchy, whichare obtained by imposing suitable second class constraints. The most drasticreduction corresponds to the (N+M)--th KdV hierarchy. Correspondingly the W(N,M) algebra is reduced to the W_{N+M} algebra. We study in detail thedispersionless limit of this hierarchy and the relevant reductions.Comment: 40 pages, LaTeX, SISSA-171/93/EP, BONN-HE-46/93, AS-IPT-49/9