Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents

TL;DR

This paper derives explicit formulas for classical W-algebras related to minimal and short nilpotent elements in simple Lie algebras, and computes the first non-trivial PDEs of associated integrable hierarchies, revealing connections to known equations like KdV.

Contribution

It provides explicit lambda-bracket formulas for affine classical W-algebras and computes the initial PDEs of generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents.

Findings

Reduction of short nilpotent equations yields Svinolupov's equations linked to Jordan algebras.

Minimal nilpotent reductions produce integrable Hamiltonian equations on 2h-3 functions.

For g=sl_2, the equations coincide with the KdV equation.

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'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.