TY - JOUR T1 - Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents JF - Communications in Mathematical Physics 331, nr. 2 (2014) 623-676 Y1 - 2014 A1 - Alberto De Sole A1 - Victor G. Kac A1 - Daniele Valeri AB - We 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. PB - SISSA UR - http://hdl.handle.net/1963/6979 N1 - 46 pages U1 - 6967 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER -