Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras
Abstract
We put the Adler-Gelfand-Dickey approach to classical W-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the KP hierarchy, together with its generalizations and reduction to the N-th KdV hierarchy, using the formal distribution calculus and the lambda-bracket formalism. We apply the Lenard-Magri 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 (non-local) bi-Poisson structures of the matrix KP and the matrix N-th KdV hierarchies, and we prove integrability of the N-th matrix KdV hierarchy.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2014
- DOI:
- 10.48550/arXiv.1401.2082
- 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)
- E-Print:
- 47 pages. In version 2 we fixed the proof of Corollary 4.15 (which is now Theorem 4.14), and we added some references