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

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.

46 pages. Corrected an error in Section 6.2 which has led to additional equations in the case of g=sl_n and its minimal nilpotent element f

References in corpus (2)

• Dirac reduction for Poisson vertex algebras
• Integrability of Dirac reduced bi-Hamiltonian equations

Cited by in corpus (7)

• Classical W-algebras for gl_N and associated integrable Hamiltonian hierarchies
• Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras
• Classical affine W-algebras and the associated integrable Hamiltonian hierarchies for classical Lie algebras
• Dirac reduction for Poisson vertex algebras
• Minimal W-superalgebras and modular representations of basic Lie superalgebras
• Integrability of Dirac reduced bi-Hamiltonian equations
• MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine $\mathcal{W}$-algebras