THE (N,M)TH KORTEWEG-DEVRIES HIERARCHY AND THE ASSOCIATED W-ALGEBRA

A differential integrable hierarchy, which is called the (N, M)th Kort eweg-de Vries (KdV) hierarchy, whose Lax operator is obtained by prope rly adding M pseudo-differential terms to the Lax operator of the Nth KdV hierarchy is discussed herein. This new hierarchy contains both th e higher KdV hierarchy and multifield representation of the Kadomtsev- Petviashvili (KP) hierarchy as subsystems and naturally appears in mul timatrix models. The N + 2M -1 coordinates or fields of this hierarchy satisfy two algebras of compatible Poisson brackets which are local a nd polynomial. Each Poisson structure generates an extended W1 + infin ity- and W(infinity)-algebras, respectively. W(N, M) is called the gen erating algebra of the extended W(infinity)-algebra. This algebra, whi ch corresponds with the second Poisson structure, shares many features of the usual W(N)-algebra. It is shown that there exist M distinct re ductions of the (N,M)th KdV hierarchy, which are obtained by imposing suitable second class constraints. The most drastic reduction correspo nds to the (N + M)th KdV hierarchy. Correspondingly the W(N,M)-algebra is reduced to the W(N+M)-algebra. The dispersionless limit of this hi erarchy and the relevant reductions are studied in detail.