The (N,M)–th KdV hierarchy and the associated W algebra

We discuss a differential integrable hierarchy, which we call the (N,M)–th KdV hierarchy, whose Lax operator is obtained by properly adding M pseudo–differential terms to the Lax operator of the N–th KdV hierarchy. This new hierarchy contains both the higher KdV hierarchy and multi– field representation of KP hierarchy as sub–systems and naturally appears in multi–matrix 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 generate an extended W1+∞ and W∞ algebra, respectively. We call W (N,M) 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. We show 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. We study in detail the dispersionless limit of this hierarchy and the relevant reductions.