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

Abstract

A differential integrable hierarchy, which is called the (N,M)th Korteweg–de Vries (KdV) hierarchy, whose Lax operator is obtained by properly adding M pseudo- differential terms to the Lax operator of the Nth KdV hierarchy is discussed herein. This new hierarchy contains both the higher KdV hierarchy and multifield representation of the Kadomtsev–Petviashvili (KP) hierarchy as subsystems and naturally appears in multimatrix models. The N+2M−1 coordinates or fields of this hierarchy satisfy two algebras of compatible Poisson brackets which are local and polynomial. Each Poisson structure generates an extended W1+∞- and W∞-algebras, respectively. W(N,M) is called the generating algebra of the extended W∞-algebra. This algebra, which corresponds with the second Poisson structure, shares many features of the usual WN-algebra. It is shown that there exist M distinct reductions of the (N,M)th KdV hierarchy, which are obtained by imposing suitable second class constraints. The most drastic reduction corresponds to the (N+M)th KdV hierarchy. Correspondingly the W(N,M)-algebra is reduced to the WN+M-algebra. The dispersionless limit of this hierarchy and the relevant reductions are studied in detail.