01209nas a2200133 4500008004100000245010300041210006900144260001000213520075500223100002100978700002000999700002001019856003601039 2014 eng d00aAdler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras0 aAdlerGelfandDickey approach to classical Walgebras within the th bSISSA3 aWe 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.1 aDe Sole, Alberto1 aKac, Victor, G.1 aValeri, Daniele uhttp://hdl.handle.net/1963/724201294nas a2200133 4500008004100000245010400041210006900145260001000214520083900224100002101063700002001084700002001104856003601124 2014 en d00aClassical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents0 aClassical Walgebras and generalized DrinfeldSokolov hierarchies bSISSA3 aWe derive explicit formulas for lambda-brackets of the affine classical
W-algebras attached to the minimal and short nilpotent elements of any simple Lie algebra g. This is used to compute explicitly the first non-trivial PDE of the corresponding intgerable generalized Drinfeld-Sokolov hierarchies. It turns out that a reduction of the equation corresponding to a short nilpotent is Svinolupov's equation attached to a simple Jordan algebra, while a reduction of the equation corresponding to a minimal nilpotent is an integrable Hamiltonian equation on 2h-3 functions, where h is the dual Coxeter number of g. In the case when g is sl_2 both these equations coincide with the KdV equation. In the case when g is not of type C_n, we associate to the minimal nilpotent element of g yet another generalized Drinfeld-Sokolov hierarchy.1 aDe Sole, Alberto1 aKac, Victor, G.1 aValeri, Daniele uhttp://hdl.handle.net/1963/697900706nas a2200133 4500008004100000245004800041210004800089260001000137520032800147100002100475700002000496700002000516856003600536 2014 en d00aDirac reduction for Poisson vertex algebras0 aDirac reduction for Poisson vertex algebras bSISSA3 aWe construct an analogue of Dirac's reduction for an arbitrary local or
non-local Poisson bracket in the general setup of non-local Poisson vertex
algebras. This leads to Dirac's reduction of an arbitrary non-local Poisson
structure. We apply this construction to an example of a generalized
Drinfeld-Sokolov hierarchy.1 aDe Sole, Alberto1 aKac, Victor, G.1 aValeri, Daniele uhttp://hdl.handle.net/1963/698000754nas a2200133 4500008004100000245006000041210005900101260001000160520035300170100002100523700002000544700002000564856003600584 2014 en d00aIntegrability of Dirac reduced bi-Hamiltonian equations0 aIntegrability of Dirac reduced biHamiltonian equations bSISSA3 aFirst, 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.1 aDe Sole, Alberto1 aKac, Victor, G.1 aValeri, Daniele uhttp://hdl.handle.net/1963/724701002nas a2200133 4500008004100000245005800041210005500099260001000154520060700164100002100771700002000792700002000812856003600832 2014 en d00aStructure of classical (finite and affine) W-algebras0 aStructure of classical finite and affine Walgebras bSISSA3 aFirst, 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.1 aDe Sole, Alberto1 aKac, Victor, G.1 aValeri, Daniele uhttp://hdl.handle.net/1963/731401065nas a2200133 4500008004100000245012700041210006900168260001300237520058400250100002100834700002000855700002000875856003600895 2013 en d00aClassical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebras0 aClassical Walgebras and generalized DrinfeldSokolov biHamiltonia bSpringer3 aWe provide a description of the Drinfeld-Sokolov Hamiltonian reduction for
the construction of classical W-algebras within the framework of Poisson vertex algebras. In this context, the gauge group action on the phase space is translated in terms of (the exponential of) a Lie conformal algebra action on the space of functions. Following the ideas of Drinfeld and Sokolov, we then establish under certain sufficient conditions the applicability of the Lenard-Magri scheme of integrability and the existence of the corresponding integrable hierarchy of bi-Hamiltonian equations.1 aDe Sole, Alberto1 aKac, Victor, G.1 aValeri, Daniele uhttp://hdl.handle.net/1963/6978