Browsing by Author "De Sole, Alberto"

Now showing 1 - 3 of 3
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    Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras
    (SISSA, 2014-01-09) De Sole, Alberto; Kac, Victor G.; Valeri, Daniele; Mathematics
    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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    Integrability of Dirac reduced bi-Hamiltonian equations
    (SISSA, 2014-01) De Sole, Alberto; Kac, Victor G.; Valeri, Daniele; Mathematics
    First, we give a brief review of the theory of the Lenard-Magri scheme for a non-local bi-Poisson structure and of the theory of Dirac reduction. These theories are used in the remainder of the paper to prove integrability of three hierarchies of bi-Hamiltonian PDE's, obtained by Dirac reduction from some generalized Drinfeld-Sokolov hierarchies.
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    Structure of classical (finite and affine) W-algebras
    (SISSA, 2014-04-02) De Sole, Alberto; Kac, Victor G.; Valeri, Daniele; Mathematics
    First, we derive an explicit formula for the Poisson bracket of the classical finite W-algebra W^{fin}(g,f), the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra g and its nilpotent element f. On the other hand, we produce an explicit set of generators and we derive an explicit formula for the Poisson vertex algebra structure of the classical affine W-algebra W(g,f). As an immediate consequence, we obtain a Poisson algebra isomorphism between W^{fin}(g,f) and the Zhu algebra of W(g,f). We also study the generalized Miura map for classical W-algebras.