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arXiv · 1401.2082

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

Alberto De Sole·Victor G. Kac·Daniele Valeri
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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.

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BibTeXRIS

Alberto De Sole, Victor G. Kac, Daniele Valeri. 2014-01-16. Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras. https://doi.org/10.1093/imrn%2Frnv017

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