Adler–Gelfand–Dickey Approach to Classical 𝒲-Algebras Within the Theory of Poisson Vertex Algebras

Alberto De Sole, Victor G. Kac, Daniele Valeri
2015 International mathematics research notices  
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
more » ... k. 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.
doi:10.1093/imrn/rnv017 fatcat:ua6wtgvhdvarzbkfzxesm66ibu