Classical W-Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

Abstract

We derive explicit formulas for λ-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 integrable 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 ˇ−3 functions, where h ˇ is the dual Coxeter number of g. In the case when g is sl2 both these equations coincide with the KdV equation. In the case when g is not of type C[subscript n], we associate to the minimal nilpotent element of g yet another generalized Drinfeld–Sokolov hierarchy.