AdlerGelfandDickey approach to classical Walgebras within the theory of Poisson vertex algebras
Abstract
We put the AdlerGelfandDickey approach to classical Walgebras in the framework of Poisson vertex algebras. We show how to recover the biPoisson structure of the KP hierarchy, together with its generalizations and reduction to the Nth KdV hierarchy, using the formal distribution calculus and the lambdabracket formalism. We apply the LenardMagri scheme to prove integrability of the corresponding hierarchies. We also give a simple proof of a theorem of Kupershmidt and Wilson in this framework. Based on this approach, we generalize all these results to the matrix case. In particular, we find (nonlocal) biPoisson structures of the matrix KP and the matrix Nth KdV hierarchies, and we prove integrability of the Nth matrix KdV hierarchy.
 Publication:

arXiv eprints
 Pub Date:
 January 2014
 arXiv:
 arXiv:1401.2082
 Bibcode:
 2014arXiv1401.2082D
 Keywords:

 Mathematical Physics;
 Mathematics  Rings and Algebras;
 Mathematics  Representation Theory;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 35Q53 (Primary) 37K10;
 17B80;
 17B69;
 37K30 (Secondary)
 EPrint:
 47 pages. In version 2 we fixed the proof of Corollary 4.15 (which is now Theorem 4.14), and we added some references