Adler-Gelfand-Dickey approach to classical W-Algebras within the theory of Poisson vertex algebras

De Sole, A., Kac, V. G. and Valeri, D. (2015) Adler-Gelfand-Dickey approach to classical W-Algebras within the theory of Poisson vertex algebras. International Mathematics Research Notices, 2015(21), pp. 11186-11235. (doi: 10.1093/imrn/rnv017)

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 Kadomtsev–Petviashvili (KP) hierarchy, together with its generalizations and reduction to the Nth Korteweg–de Vries (KdV) hierarchy, using the formal distribution calculus and the λ-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 (nonlocal) bi-Poisson structures of the matrix KP and the matrix Nth KdV hierarchies, and we prove integrability of the Nth matrix KdV hierarchy.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Valeri, Dr Daniele
• Authors: De Sole, A., Kac, V. G., and Valeri, D.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: International Mathematics Research Notices
• Publisher: Oxford University Press
• ISSN: 1073-7928
• ISSN (Online): 1687-0247
• Published Online: 09 February 2015