Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1993-11-11
J.Math.Phys. 35 (1994) 5781-5819
Physics
High Energy Physics
High Energy Physics - Theory
40 pages, LaTeX, SISSA-171/93/EP, BONN-HE-46/93, AS-IPT-49/93
Scientific paper
10.1063/1.530710
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 {\it local} and {\it polynomial}. Each Poisson structure generate an extended W_{1+\infty} and W_\infty algebra, respectively. We call W(N, M) the generating algebra of the extended W_\infty algebra. This algebra, which corresponds with the second Poisson structure, shares many features of the usual $W_N$ 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 W_{N+M} algebra. We study in detail the dispersionless limit of this hierarchy and the relevant reductions.
Bonora Loriano
Xiong Chuan-Sheng
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