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/1963/6979