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

L. Bonora; C. S. Xiong

doi:10.1063/1.530710

paper

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

arXiv:hep-th/9311070 · doi:10.1063/1.530710

Abstract

We discuss a differential integrable hierarchy, which we call the (N, M) $- - t h K d V hi er a r c h y, w h ose L a x o p er a t or i so b t ain e d b y p r o p er l y a dd in g$ M $p se u d o - - d i f f er e n t ia l t er m s t o t h e L a x o p er a t or o f t h e N - - t h K d V hi er a r c h y . T hi s n e w hi er a r c h y co n t ain s b o t h t h e hi g h er K d V hi er a r c h y an d m u l t i - - f i e l d r e p r ese n t a t i o n o f K P hi er a r c h y a ss u b - - sy s t e m s an d na t u r a l l y a pp e a r s inm u l t i - - ma t r i x m o d e l s . T h e N + 2 M - 1 coor d ina t esor f i e l d so f t hi s hi er a r c h y s a t i s f y tw o a l g e b r a so f co m p a t ib l e P o i sso nb r a c k e t s w hi c ha r e local an d polynomial . E a c h P o i sso n s t r u c t u r e g e n er a t e an e x t e n d e d W_{1 + \infty} an d W_{\infty} a l g e b r a, r es p ec t i v e l y . W ec a l l W (N, M) t h e g e n er a t in g a l g e b r a o f t h ee x t e n d e d W_{\infty} a l g e b r a . T hi s a l g e b r a, w hi c h cor r es p o n d s w i t h t h eseco n d P o i sso n s t r u c t u r e, s ha r es man y f e a t u r eso f t h e u s u a l$ 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.

40 pages, LaTeX, SISSA-171/93/EP, BONN-HE-46/93, AS-IPT-49/93