@article {2013,
title = {Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents},
journal = {Communications in Mathematical Physics 331, nr. 2 (2014) 623-676},
number = {arXiv:1306.1684;},
year = {2014},
note = {46 pages},
publisher = {SISSA},
abstract = {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{\textquoteright}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.},
doi = {10.1007/s00220-014-2049-2},
url = {http://hdl.handle.net/1963/6979},
author = {Alberto De Sole and Victor G. Kac and Daniele Valeri}
}