01294nas 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/1963/697901065nas 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