Classical ${\mathcal{W}}$-Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

Abstract

We derive explicit formulas for λ-brackets of the affine classical ${\mathcal{W}}$ -algebras attached to the minimal and short nilpotent elements of any simple Lie algebra ${\mathfrak{g}}$ . 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 $\mathfrak{g}$ . In the case when $\mathfrak{g}$ is ${\mathfrak{sl}_2}$ both these equations coincide with the KdV equation. In the case when $\mathfrak{g}$ is not of type ${C_n}$ , we associate to the minimal nilpotent element of $\mathfrak{g}$ yet another generalized Drinfeld–Sokolov hierarchy.