Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras
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Date
2014-01-09
Journal Title
Journal ISSN
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Publisher
SISSA
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.
Description
45 pages